Conference Paper
2022

Dynamic Transition From Regular to Mach Reflection Over a Moving Wedge

Lubna Margha1,3, Ahmed A. Hamada1, Doyle D. Knight2, Ahmed Eltaweel4
1Department of Ocean Engineering, Texas A$\&$M University, College Station,
TX, 77843, USA
2Department of Mechanical and Aerospace Engineering, Rutgers - The State
University of New Jersey, Piscataway, NJ, 08854, USA
3Department of Aeronautical and Aerospace Engineering, Cairo University, Giza
12613, Egypt
4Aerospace Engineering Program, University of Science and Technology - Zewail
City, Giza, 12578, Egypt

## Abstract

The transition between the Regular Reflection (RR) and Mach Reflection (MR)
phenomenon impacts the design of the supersonic and hypersonic air-breathing
vehicles. The aim of this paper is to numerically investigate the dynamic
transition from RR to MR of unsteady supersonic flow over a two-dimensional
wedge, whose trailing edge moves along the $x$-direction upstream with a
velocity, $V(t)$ at a free-stream Mach number of $M_{\infty}=3$. The
simulation is conducted using the unsteady compressible inviscid flow solver,
which is implemented in OpenFOAM®, the open-source CFD tool. Further, the
wedge motion is applied by moving the mesh boundary, performing the Arbitrary
Lagrangian-Eulerian (ALE) technique. In addition, the sonic and detachment
criteria are used to define the dynamic transition from RR to MR during the
increase of the wedge angle. Different reduced frequencies, $\kappa$, in the
range of $[0.1-2]$ for the moving wedge are applied to study the lag in the
dynamic transition from the steady-state condition. The results show that the
critical value of $\kappa=0.4$ distinguishes between the rapid and gradual lag
in the transition from RR to MR. In addition, the transition from RR to MR
occurs above the Dual Solution Domain (DSD), since the shock is curved
downstream during the rapid motion of the wedge.

Keywords: Regular reflection; Mach reflection; Moving wedge; Dynamic shock
waves; Supersonic flow; Dual solution domain.

## 1\. Introduction

Predicting the shock reflections and the shock wave interactions are very
crucial in the design and operation phases of many engineering applications,
such as shock-wave focusing, protection against blasts and detonations, and
supersonic and hypersonic vehicles. For instance, a series of shock wave
interactions are generated in the scramjet inlet, using tilted wedges, to
decelerate the flow and achieve efficacious combustion. A proper comprehension
of dynamic shock wave interactions over simple moving geometries, such as a
wedge, will give insights into the limitations of the moving supersonic intake
for an efficient operation.

When a supersonic flow impinges a wedge, an incident shock is generated. Then,
it reflects on the mid-plane of symmetry, creating a second shock. Figure 1
shows that the reflection of the incident shock on the mid-plane will follow
one of two configurations, Regular Reflection (RR) or Mach Reflection (MR),
which depends on the free-stream Mach number, $M_{\infty}$, and the incident
shock angle, $\beta$, [1, 2]. The structure of RR is formed of two shock
waves, the incident shock wave (I), and the reflected shock wave (R), as shown
in Figure 1(a). They gather on the reflecting surface at a point, called the
reflection point (RP). The reflected shock deforms slightly from a straight
line due to the interference with the Expansion fans (E), generated at the
trailing edge of the wedge. On the other hand, the structure of the MR
includes three shock waves, the incident shock wave (I),

the reflected shock wave (R), and the Mach stem (MS), which all meet at the
triple point (TP) and a slip line (S) appears, as shown in Figure 1(b). The
slip line is formed due to the difference in the flow parameters behind the
reflected shock and the Mach stem. The subsonic region behind the Mach stem
bounded by the slip lines and the sonic throat (ST), is called the subsonic
pocket (SP). Again, expansion fans (E), interfere with the reflected shock and
bend it. The weak waves, that propagate behind the reflected shock from
expansion waves, reach the slip line causing the generation of Kelvin-
Helmholtz vortices (KHV). The transition from RR to MR over a wedge occurs
when the wedge angle is large enough, that the reflected shock is no longer
able to turn the flow parallel to the mid-plane. Thus, a normal shock is
created and MR happens.

(a) RR configuration

(b) MR configuration

Figure 1:  The shock structure of the RR and MR configurations, respectively.

John von Neumann [3] is the first one who introduced two different criteria
for the transition between RR and MR for symmetrical reflection, known as the
detachment criterion and the von Neumann (or mechanical equilibrium)
criterion. The detachment criterion is used to determine the transition from
RR to MR and denotes the maximum shock deflection angle $\beta_{D}$ that a RR
configuration is theoretically possible. At a higher angle, the reflection
point is forced to detach from the mid-plane forming the Mach stem. The von
Neumann criterion denotes the theoretical limit of the shock deflection angle
$\beta_{N}$ for the MR configuration. Subsequently, the length scale criterion
is proposed by Hornung et al. [4]. This criterion assumes the MR occurs at a
certain length of the Mach stem. Moreover, the sonic criterion is also used to
define the transition from RR to MR, when the flow beyond the reflected shock
becomes sonic which assures the beginning of the Mach stem. Thus, the
detachment criterion and the sonic criterion are very close. Furthermore, Both
the RR and MR are possible solutions to exist in a range of wave angles
($\beta_{N}<\beta<\beta_{D}$) at relatively high Mach numbers
($M_{\infty}>2.2$ for a perfect gas with the specific heat ratio
$\gamma=1.4$), which called Dual Solution Domain (DSD) [5]. The supposition of
hysteresis within the DSD in steady high-speed flows was first assumed by
Hornung et al. [4]. Then, they tried to confirm this hypothesis experimentally
[5], but they observed no hysteresis because of the disturbances in the wind
tunnel. After that, Chpoun et al. [6] well studied the hysteresis phenomenon
experimentally. In addition, Ivanov et al. used different numerical methods,
such as Direct Simulation Monte Carlo [7], and Euler approach [8], and also
used the kinetic and continuum approaches [9], to carefully investigate the
hysteresis.
The transition between RR and MR within the DSD can be achieved using the
pulsed energy deposition [10, 11], or the movement of the wedge [12]. Felthun
and Skews [13] simulated the dynamic transition between RR and MR over a
rotating wedge around its leading edge at $M_{\infty}=3$ and various rates.
They found that the nature of the transition is dependent on the rate of the
wedge rotation. At a very low trailing-edge Mach number ($M_{t}=0.001$), the
transition occurs near the theoretical limit. On the other hand, when the
wedge angle changes at higher rates, the transition angles are beyond the DSD.
Further, the lag between the dynamic transition angle, $\beta_{t}$, from RR to
MR and the transition wave angle in the static case
($\beta_{t_{SC}}=39.5^{o}$) increases rapidly at relatively low $M_{t}$ to an
asymptotic value around $42^{o}$ at $M_{t}>0.05$. Later on, Naidoo and Skews
[14] continued the work using a rapidly rotating wedge and studied the
transition between RR and MR in the weak and strong-reflection ranges. Their
experimental and numerical results showed that the transition from RR to MR
happens beyond the steady-state detachment criterion, and the transition from
MR to RR occurs below the von Neumann theoretical limit. Furthermore, Ivanov
et al. [8] conducted numerical and experimental investigations over a rotating
wedge around its trailing edge in a supersonic flow with $M_{\infty}=4$. The
numerical simulations were used to study the hysteresis. In addition, their
experiments showed that the transition between RR and MR occurs around the von
Neumann angle.
Numerical simulations were conducted in the present work to investigate the
dynamic transition from RR to MR using a two-dimensional wedge at a free-
stream Mach number $M_{\infty}=3$. The trailing edge of the wedge moves
horizontally upstream (the wedge height is kept constant) with a constant
reduced frequency, $\kappa$. Further, different values of $\kappa$ were tested
to study the lag in the transition, and the development of the Mach stem
height.

## 2\. Physical Model

### 2.1 Model Description

The flow configuration of two symmetrical wedges in a supersonic flow with a
free-stream Mach number, $M_{\infty}=3$, is shown in Figure 2. The height of
the computational inflow boundary is $2H$, the height of each wedge is $h$,
the initial wedge chord is $w(0)=1$, the stream-wise length of the wedge is
$L(t)$, the length of the wedge plus flat plate is $L_{t}$, and the time-
dependent wedge angle is $\theta(t)$. The trailing edge of the wedge moves in
the $x$-direction with velocity, $V(t)$, from its initial location. Thus, the
wedge angle, $\theta(t)$, and wedge length, $L(t)$, change during the motion,
while $H$, $h$ and $L_{t}$ remain constant. The flow is assumed two-
dimensional and inviscid, and the gas is calorically and thermally perfect.
When the supersonic flow incident on a wedge, the wedge angle generates an
oblique shock wave that is reflected on the plane of symmetry. According to
the angle of the wedge, there are two scenarios for the reflected shock;
either a RR configuration or a MR configuration is formed. The dynamic
transition from RR to MR during the increase of $\theta(t)$ from $19^{\circ}$
to $34^{\circ}$ is studied for different constant values of $\kappa$ at a
range of $[0.1,2]$ with step $0.1$. Due to the lag in the shock system, there
are two wave angles that can be measured; wave angle at reflection$/$triple
point, $\beta_{p}$, and tangent wave angle at wedge’s apex, $\beta_{tang}$.
Table 1 shows the values of these parameters.

Figure 2:  The flow configuration of the horizontal motion of two symmetrical wedges in a supersonic flow. Table 1: System properties and parameters. Initial wedge’s chord, $w(0)$ | $1m$
---|---
Initial wedge angle, $\theta(0)$ | $19^{\circ}$
Final wedge angle, $\theta(t_{f})$ | $34^{\circ}$
Wedge height to half domain height, $\frac{h}{H}$ | $0.3617$
Initial wedge length to half domain height, $\frac{L(0)}{H}$ | $1.0506$
Total wedge length to half domain height, $\frac{L_{t}}{H}$ | $2$
Free-stream Mach number, $M_{\infty}$ | $3$
Reduce frequency, $\kappa$ | $0.1:0.1:2$

### 2.2 Governing Equations

Two-dimensional unsteady compressible Euler equations are used to model the
supersonic flow over a wedge and are expressed in the conservative form as:

$\frac{\partial Q}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial
G}{\partial y}=0,$ (1)

where

$Q=\begin{bmatrix}\rho\\\ \rho u\\\ \rho v\\\ \rho e\end{bmatrix},\quad
F=\begin{bmatrix}\rho u\\\ \rho u^{2}+p\\\ \rho uv\\\ u(\rho
e+p)\end{bmatrix},\quad G=\begin{bmatrix}\rho v\\\ \rho uv\\\ \rho v^{2}+p\\\
v(\rho e+p)\end{bmatrix}$ (2)

The static pressure is obtained from

$p=(\gamma-1)\left(\rho e-\rho\frac{u^{2}+v^{2}}{2}\right)$ (3)

where $u$ and $v$ are the velocity components in the Cartesian coordinates $x$
and $y$, respectively, $\rho$, $p$ and $e$ are the density, the pressure, and
the internal energy of the flow field, respectively, and $\gamma$ is the
specific heat ratio of air, which equals $1.4$.

### 2.3 Equations of Motion

The trailing edge of the wedge moves horizontally with velocity $V(t)$ from
its initial location with a constant wedge angular velocity,
$\omega=d\theta/dt$ ($sec^{-1}$). Because the wedge height, $h$, is kept
constant, the increase in the wedge angle, $\theta(t)$, changes only the wedge
stream-wise length, $L(t)$, and they are expressed as:

$L(t)=h\ cot\left(\theta(t)\right)$ (4a)
$\theta(t)=tan^{-1}\left(\frac{h}{L(0)}\right)+\omega t$ (4b)

where $L(0)$ is the initial wedge stream-wise length at the starting time,
$t=0$, of the motion.
Further, the dimensional velocity of the trailing edge of the wedge is
expressed as:

$V_{t}(t)=\omega\ h\sqrt{1+cot^{2}\left(\theta(t)\right)}$ (5)

In addition, the wedge angular velocity, $\omega$, and the time of the motion,
$t$, are normalized using the free-stream velocity and the initial wedge
stream-wise length at time $t=0$, $L(0)$, to introduce the non-dimensional
frequency, $\kappa$, and the non-dimensional time, $\tau$, which are defined
as:

$\kappa=\frac{\omega\ L(0)}{U_{\infty}}$ (6a) $\tau=\frac{t\
U_{\infty}}{L(0)}$ (6b)

Therefore, the trailing edge Mach number can be written as:

$M_{t}(\tau)=\kappa\ M_{\infty}\
tan\left(\theta(0)\right)\sqrt{1+cot^{2}\left(\theta(\tau)\right)}$ (7)

where $\theta(\tau)$, the wedge angle as a function of non-dimensional time,
is defined as:

$\theta(\tau)=\theta(0)+\kappa\tau$ (8)

where $\theta(0)$ is the initial wedge angle at the starting non-dimensional
time, $\tau=0$, of the motion.

## 3\. Computational Model

### 3.1 Numerical Implementation

rhoCentralDyMFoam is a transient, density-based compressible flow solver with
support for a mesh-motion, that is implemented in OpenFOAM®$-$v2006. The
solver’s technique combines the semi-discrete and upwind-central non-staggered
schemes of Kurganov and Tadmor [15, 16]. The previously mentioned schemes are
implemented in the solver in order to avoid the use of Riemann solvers of
characteristic decomposition [17]. In addition, rhoCentralDyMFoam depends on
an operator-splitting method to solve both momentum and energy equations.
Further, explicit predictor equations are implemented for the convection of
the conserved variables, whereas implicit corrector equations are used for the
diffusion of the primitive variables [17]. The van Leer limiter, conducted in
the rhoCentralDyMFoam, efficiently balances the performance of the solution
through shock capture, oscillations-free fields, and computational cost [18].

### 3.2 Computational Domain

Since the problem is symmetric in the geometry and the flow behavior, only
half of the domain is computed. The computational domain is body-fitted for
the wedge with an initial wedge angle of $19^{\circ}$, as shown in Figure 3.
The streamwise length and the transverse length of the domain are kept
constant during the simulations of $2.2w(0)$ and $0.9w(0)$, respectively,
where $w(0)$ is the initial wedge chord length at $t=0$. The origin is placed
at the leading edge point of the wedge, and the total length of the wedge,
$L_{t}$, is $1.8w(0)$. The domain is bounded by four boundaries: inlet,
outlet, bottom, and top. The flow enters the domain with supersonic Mach
number, $M_{\infty}=3$, and free-stream pressure and temperature at sea level.
At the outlet boundary, a zero gradient boundary condition is applied to all
of the flow variables, in order to set the outlet field to the internal field
value. The bottom boundary condition (the wedge and the flat plate in front of
the wedge) has a slip velocity boundary condition with a zero gradient
boundary condition for the pressure and temperature. The symmetric plane
boundary conditions are applied at the top surface.

Figure 3: Schematic of the computational domain, the boundary, and initial
conditions, for mesh 1

### 3.3 Grid Generation

In this study, the delay in the dynamic transition from the static case value
is mainly caused by two reasons; the mesh resolution and the rate of motion of
the wedge trailing edge. the moving rate will be discussed further in the
results section. Hence, the mesh-independent test is very critical for more
accurate results. A 2D-ordered grid of quadrilateral elements was used to
discretize the computational domain. The domain was divided into 9 blocks as
shown in Figure 3. Orthogonal grids were generated in the blocks near the
wedge surface and the mid-plane of symmetry, to reduce the computational noise
on the results. Moreover, the middle blocks’ edges were curved to improve the
grid orthogonality and maintain the same aspect ratio in that region.

In order to accurately capture the transition point from RR to MR and the Mach
stem height, a mesh-independent study was performed. Different mesh sizes were
implemented at the free-stream Mach number of 3, a reduced frequency
$\kappa=0.726$. The study was started with mesh 1, of $328\times 90$ elements.
Then, the mesh size was doubled to generate mesh 2, mesh 4, mesh 8, and mesh
16. The time step was allowed to change during each simulation, but it was
controlled with an upper limit of the Courant–Friedrichs–Lewy (CFL) number,
which was set to be $0.2$. Figure 4 indicates the variation of the Mach number
along the mid-plane of symmetry ($y=0.9$) at the sonic criterion
($M_{transition\ point}\cong 1$). It shows the convergence of the results with
the increase in the mesh size. Moreover, the

Figure 4: Independent mesh study: The Mach number variation along the top symmetry plane at $y=0.9$, and at the sonic criterion of transition, using different grid sizes. Table 2: Independent Mesh test: Absolute percentage error of the transition wave angle at the reflection point, $\beta_{t_{p}}$, and the Mach stem height, MS, at wedge angle, $\theta$=$27^{\circ}$. Mesh # | Mesh Size | $\beta_{t_{p}}\ (^{\circ})$ | $\frac{\mathrm{MS}}{L(0)}\times 10^{-2}$ | $|Error|\%$
---|---|---|---|---
$\beta_{t_{p}}$ | $\frac{\mathrm{MS}}{L(0)}$
1 | $328\ \times 90\ \ $ | $41.39$ | 5.29 | $1.22$ | 46.4
2 | $656\ \times 180\ $ | $41.15$ | 7.34 | $0.62$ | 25.6
4 | $1312\times 360\ $ | $41.03$ | 8.75 | $0.33$ | 11.3
8 | $2624\times 720\ $ | $40.96$ | 9.53 | $0.17$ | 3.4
16 | $5248\times 1440$ | $40.89$ | 9.87 | - | -

deviation between mesh 8 and mesh 16 in the Mach distribution can be
neglected. Further, Table 2 compares different mesh sizes, showing the
absolute percentage error of the transition wave angle at the reflection
point, $\beta_{t}$, and the Mach stem height, MS, at wedge angle,
$\theta=27^{\circ}$. It is clear that for low mesh quality ($328\times 90$),
the error percentage of developing the Mach stem height, MS, at wedge angle of
$\theta=27^{\circ}$, is around $50\%$, which is a huge error that would be a
miss-leading in determining the dynamic transition point. Moreover, this error
percentage rapidly declined by doubling the mesh size. Therefore, both mesh 8
and mesh 16 provide more precise results. In order to reduce the computational
time, mesh 8 is selected to implement all the simulations. The minimum element
size in mesh 8 is $0.75mm\times 0.78mm$, and the time step to ensure the CFL
number of $0.2$ is $93.5ns$.

## 4\. Verification

Figure 5: Pressure contours at different wedge angles and $M_{t}=0.05$. The
present work is shown in the colored left sub-figures, whereas Felthun and
Skews [13] work is shown in the right sub-figures.

The inserted dynamic code into the OpenFOAM solver is verified using the
supersonic flow over a dynamic rotating wedge problem. The angle of a unity
chord wedge is varied with different trailing-edge Mach numbers, $M_{t}$, in a
flow of a free-stream Mach number of $3$. The observed transition angles,
$\beta_{t}$, from regular to Mach reflection are obtained for these wedge
rotation simulations. Figure 5 shows the comparison of the pressure contours
for the flow over the rotating wedge with $M_{t}=0.05$ at different wedge
angles, $\theta$. Further, Figure 6 indicates that our work is within the
margin of accuracy of measurements for the work of Felthun and Skews [13].

Figure 6: Validation with the work of Felthun and Skews [13] by measuring the
effect of trailing-edge Mach number on transition angles from regular to Mach
reflection

## 5\. Results and Discussion

The dynamic transition from RR to MR was studied using a horizontal moving
wedge at a $M_{\infty}=3$ and was modeled by starting with RR in a steady flow
with a wedge angle of $19^{\circ}$ and increasing it to $34^{\circ}$. The
sonic and Detachment criteria were used to determine the transition point.
Different reduced frequencies $(0.1:0.1:2)$ were tested to indicate the lag
effect in the transition angles, $\theta_{t}$ and $\beta_{t}$, and the Mach
stem height, MS. The used system properties and parameters are listed in Table
1.

### 5.1 Dynamic Transition from RR to MR

The sudden motion of the wedge is started horizontally from the steady-state
at $\theta=19^{\circ}$ with different reduced frequencies, $\kappa$. This
movement affects the shock system. A lag appears in the incident shock wave
angle behind the steady-state value at the same wedge angle, causing a
curvature in the incident shock, which is indicated with the tangent wave
angle, $\beta_{tang}$. Further, the strength of these effects depends on the
value of $\kappa$. The dynamic transition from RR to MR is defined according
to the sonic and detachment criteria. Figure 7 is a close view of two
transition criteria over the used mesh at a $\kappa=0.1$. The temporal time of
the sonic criterion is determined when there is a point with a Mach number of
$1$ on the top surface, as shown in Figure 7 (a). After a non-dimensional time
of $3.46$ for the case of $\kappa=0.1$, a sharp bend in the reflected shock
wave happens, detaching the shock from the symmetric surface and generating
the triple point. This would decrease the Mach number below $1$ and when the
Mach number reaches $0.475$ (Mach number after a normal shock) in a point on
the symmetric plane, the detachment criterion is full-filled, as shown in
Figure 7 (b). Further, in between the sonic and detachment limits, a Regular
Reflection with Subsonic Downstream Flow (RRs) occurs instead of Regular
Reflection (RR). In addition, Figure 14 shows the lag and curvature effects in
the system with $\kappa=0.5$, where the transition happens at a wedge angle at
the sonic limit of $\theta_{t_{SL}}=23.15^{\circ}\pm 0.1^{\circ}$ and at the
detachment limit at $\theta_{t_{DL}}=23.87^{\circ}\pm 0.1^{\circ}$, while the
transition of the static case occurs at a wedge angle
$\theta_{t_{SC}}=21.5^{\circ}$.

Figure 7: Density contours when the dynamic transition from RR to MR occurs
with $\kappa=0.1$ at: (a) Sonic and (b) Detachment criteria.

The investigation of the transition from RR to MR was conducted by studying
the change of the transition non-dimensional times, $\tau_{t}$, transition
wedge angles, $\theta_{t}$, and transition wave angles, $\beta_{t}$, at the
two criteria with various $\kappa$. The increase in reduced frequency
represents an increase in the velocity of the moving wedge. Thus, the required
time to change the state will decrease with the increase of $\kappa$. After
the significant decrease in the transition non-dimensional time, $\tau_{t}$ at
low values of $\kappa$, it starts to gradually decrease at higher values of
$\kappa$ than $0.4$, reaching $\tau_{t}=4$ from the starting wedge angle of
$\theta=19^{\circ}$ at $\kappa=2$, as shown in Figure 8 right. Further, the
change in the non-dimensional time

Figure 8: The variation of the transition wedge angle, $\theta_{t}$, and the
transition non-dimensional time, $\tau_{t}$, with various reduced frequencies,
$\kappa$ at the Sonic limit, SL, and the Detachment limit, DL. Gray straight
lines are plotted to indicate the non-linear effect of the lag.

between the detachment and sonic limits decreases, dramatically at low values
of $\kappa$ and slightly at high values of $\kappa$, from $3.46$ in the case
of $\kappa=0.1$ to $0.95$ in the case of $\kappa=2$. In addition, the lag in
the flow appears with the wedge motion. Thus, there will be time needed to
transfer the information from the wedge apex to the end of the incident shock.
During this time, the wedge continues its motion. That is why the transition
wedge angle in the dynamic cases is larger than that of the static case. This
is shown in Figure 8 left with a Gauss curve fit and the uncertainty in
measuring the angles is $\pm 0.1^{\circ}$. Consequently, increasing the motion
rate of the wedge, $\kappa$, increases the gap between the dynamic and the
static transition wedge angles. Hence, the reduced frequency generates a
spatial and temporal lag in the system.
The difference between the tangent transient wave angle, $\beta_{t_{tang}}$,
and the wave angle at reflection$/$triple point, $\beta_{t_{p}}$, shown in
Figure 9, emphasizes the lag in the propagation of the information along the
incident shock wave. Even at a very low value of $\kappa$, such as
$\kappa=0.1$, the lag in $\beta_{t_{tang}}$ from the static case (about
$2.5^{\circ}\pm 0.1^{\circ}$), which represents the curvature of the incident
shock at the apex of the wedge, is relatively large with respect to the lag in
$\beta_{t_{p}}$ (about $0.5^{\circ}\pm 0.1^{\circ}$). At the values of
$\kappa$ smaller than $0.4$, the differences in wave angles, $\beta_{t_{p}}$
and $\beta_{t_{tang}}$, between the sonic and detachment criteria are within a
fraction of a degree. Further, the differences increase gradually and reach
$2.5^{\circ}\pm 0.1^{\circ}$ for $\beta_{t_{p}}$ and $3.2^{\circ}\pm
0.1^{\circ}$ for $\beta_{t_{tang}}$ at $\kappa=2$.
Another way to study the lag in the shock wave system is achieved with the
ratio of the difference between the wave angles in the dynamic and static
cases to the difference between the wedge angles in the dynamic and static
cases, $(\beta_{t}-\beta_{t_{SC}})/(\theta_{t}-\theta_{t_{SC}})$, as shown in
Figure 10. This ratio represents the ratio between the speed of information
propagation along the incident shock wave to the speed of wedge motion. The
ratio decreases steeply at $\kappa$ smaller than 0.4; i.e. the speed of
information propagation decreases dramatically with the increase of $\kappa$
at the low-value range. On the other hand, the ratio decreases almost linearly
with a lower negative slope at the value of $\kappa$ larger than $0.4$.

Figure 9: The variation of transition wave angle, $\beta_{t}$, with various
reduced frequencies, $\kappa$ at the Sonic limit, SL, and the Detachment
limit, DL. Gray straight lines are plotted to indicate the non-linear effect
of the lag.  Figure 10: The variation of the lag in the transition wave angle
to the lag in the transition wedge angle ratio,
$(\beta_{t}-\beta_{t_{SC}})/(\theta_{t}-\theta_{t_{SC}})$, with various
reduced frequencies, $\kappa$ at the Sonic limit, SL, and the Detachment
limit, DL.

### 5.2 Mach Stem Height

In order to examine the effect of changing the movement rate of the wedge on
the development of the Mach stem, MS, its height, $\textrm{MS}/L(0)$, was
measured at a certain wedge angle, $\theta=27^{\circ}$, for different values
of $\kappa$, as shown in Figure 15. At the static case, a MR happens at wedge
angle, $\theta_{SC}$, of $27^{\circ}$ with a $\textrm{MS}_{SC}/L(0)=0.308$. On
the other hand, the dynamic Mach stem height decreases with the increase of
$\kappa$. At a very high value of $\kappa$, the $\textrm{MS}/L(0)$ disappears
and the RR occurs, due to the excessive lag in transition, as shown in Figure
15. Moreover, Figure 11 shows that longer time scales (small values of
$\kappa$) enable the Mach stem to develop close to the static case value. For
example, the $\textrm{MS}/L(0)$ is about $99.4\%$ of that of the static case
at $\kappa=0.1$. Further, the deviation of the dynamic Mach stem height from
the static case, $\textrm{MS}_{SC}/L(0)-\textrm{MS}/L(0)$, increases gradually
with the increase of $\kappa$, reaching a percentage value of $98.4\%$ with
respect to the static case at $\kappa=1.3$.

Figure 11: The variation of non-Dimensional Mach stem height and the
corresponding Non-Dimensional time at wedge angle, $\theta=27^{\circ}$, with
various reduced frequencies, $\kappa$.

### 5.3 Shock Reflection Domain

Theoretically, when a supersonic flow impinges a tilted

Figure 12: Comparison of dynamic transition wave angle, $\beta_{t_{tang}}$
with the analytical steady transition wave angle, $\beta_{t_{sc}}$. Figure 13:
Mach contours at wedge angle of $\theta=34^{\circ}$, at different reduced
frequencies, (a) $\kappa=1.5$ and (b) $\kappa=2.0$. The subsonic and
supersonic regions are represented by white and black contours, respectively.

wedge, there are many possible shock wave systems, such as Regular Reflection
(RR), Mach Reflection (MR), Mach Reflection with Subsonic Downstream Flow
(MRs), Inverted Mach Reflection (IMR), etc, as mentioned by Mouton [19]. The
generation of a specific configuration depends on the free-stream Mach number
and the wedge angle$/$wave angle. Figure 12 shows the physical limits of each
shock configuration for $\gamma=1.4$, including the dynamic tangent transition
wave angles, $\beta_{t_{tang}}$, at a free-stream Mach number of $3$ and
different values of $\kappa$. The horizontal upstream motion of the wedge
(increasing the wedge angle) generates a lag in $\beta_{t_{tang}}$, placing it
above the DSD. Furthermore, increasing the reduced frequency moves the
transition angle far away from the DSD. At high values of $\kappa$,
$\beta_{t_{tang}}$ becomes very close to the sonic reflected shock condition
(MRs). This was indicated by a small triangular subsonic zone in the
reflection domain between the slip-line and the reflected shock at
$\kappa=1.5$ and $2$ and $\theta=34^{\circ}$, as shown in Figure 13. At
$\kappa=2$, the triangular subsonic zone is very small due to the excessive
lag in the shock system, which delays the development of MS. Thus, it became
very close to the expansion fan that quickly accelerates the flow to
supersonic speeds.

## 6\. Conclusion

The current research work aims to investigate the dynamic transition from RR
to MR over a two-dimensional slip wedge at a free-stream Mach number of 3. The
trailing edge of the wedge moves horizontally upstream with various reduced
frequencies, $\kappa=0.1:0.1:2$. The phenomenon was studied by analyzing the
variation of the transition angles, and examining the lag in the development
of the Mach stem with various reduced frequencies. Further, a comparison
between the dynamic and theoretical static transition was achieved. The major
conclusions of the study are:

  * •

The time scale needed to reach the transition declines rapidly at low values
of $\kappa$, and slowly with high values of $\kappa$. However, the lag in the
transition parameters, wedge angle, and wave angles, changes gradually with
$\kappa$.

  * •

The ratio of the speed of information propagation along the incident wave to
the speed of wedge motion falls quickly at lower values of $\kappa$ than $0.4$
and decreases with an almost constant lower rate at higher values.

  * •

Although the difference between the theoretical sonic and detachment
conditions is within a degree, the difference gradually increases higher than
a degree for $\kappa>0.4$.

  * •

The growth of the Mach stem is very sensitive to the numerical resolution and
the lag due to the motion of the wedge with a reduced frequency, $\kappa$.

  * •

At low values of $\kappa$, the dynamic transition occurs slightly above the
theoretical detachment limit (MR). On the opposite side, the dynamic
transition wave angle tends to reach the physical limit of the Mach Reflection
with Subsonic Downstream Flow (MRs) at high values of $\kappa$.

## References

  * Hornung [1986] Hans Hornung.  Regular and mach reflection of shock waves.  _Annual Review of Fluid Mechanics_ , 18(1):33–58, 1986.
  * Ben-Dor [2007] Gabi Ben-Dor.  _Shock Wave Reflection Phenomena_ , volume 2.  Springer, 2007.
  * von Neumann [1943] John von Neumann.  Oblique reflection of shocks.  _Bureau of Ordinance, Explosives Research Report_ , 1943.
  * Hornung et al. [1979] HG Hornung, H Oertel, and RJ Sandeman.  Transition to mach reflexion of shock waves in steady and pseudosteady flow with and without relaxation.  _Journal of Fluid Mechanics_ , 90(3):541–560, 1979.
  * Hornung and Robinson [1982] HG Hornung and ML Robinson.  Transition from regular to mach reflection of shock waves part 2. the steady-flow criterion.  _Journal of Fluid Mechanics_ , 123:155–164, 1982.
  * Chpoun et al. [1995] A Chpoun, D Passerel, H Li, and G Ben-Dor.  Reconsideration of oblique shock wave reflections in steady flows. part 1. experimental investigation.  _Journal of Fluid Mechanics_ , 301:19–35, 1995.
  * Ivanov et al. [1995] MS Ivanov, SF Gimelshein, and AE Beylich.  Hysteresis effect in stationary reflection of shock waves.  _Physics of Fluids_ , 7(4):685–687, 1995.
  * Ivanov et al. [2001] Mikhail S Ivanov, D Vandromme, VM Fomin, AN Kudryavtsev, A Hadjadj, and DV Khotyanovsky.  Transition between regular and mach reflection of shock waves: new numerical and experimental results.  _Shock Waves_ , 11(3):199–207, 2001.
  * Ivanov et al. [1998] MS Ivanov, GN Markelov, AN Kudryavtsev, and SF Gimelshein.  Numerical analysis of shock wave reflection transition in steady flows.  _AIAA Journal_ , 36(11):2079–2086, 1998.
  * Yan et al. [2003] H Yan, R Adelgren, G Elliott, Doyle Knight, and T Beutner.  Effect of energy addition on mr $\backslash\rightarrow$ rr transition.  _Shock Waves_ , 13(2):113–121, 2003.
  * Khotyanovsky et al. [2004] D Khotyanovsky, Doyle Knight, A Kudryavtsev, and M Ivanov.  Parallel simulation of laser-induced shock wave reflection transition.  In _Parallel Computational Fluid Dynamics 2003_ , pages 365–372. Elsevier, 2004.
  * Felthun and Skews [2002] L Felthun and B Skews.  Transition between regular and mach reflection in dynamic compressible flows.  In _40th AIAA Aerospace Sciences Meeting & Exhibit_, page 549, 2002\.
  * Felthun and Skews [2004] Luke T Felthun and Beric W Skews.  Dynamic shock wave reflection.  _AIAA Journal_ , 42(8):1633–1639, 2004.
  * Naidoo and Skews [2011] Kavendra Naidoo and Beric W Skews.  Dynamic effects on the transition between two-dimensional regular and mach reflection of shock waves in an ideal, steady supersonic free stream.  _Journal of Fluid Mechanics_ , 676:432–460, 2011.
  * Kurganov and Tadmor [2000] Alexander Kurganov and Eitan Tadmor.  New high-resolution semi-discrete central schemes for hamilton–jacobi equations.  _Journal of Computational Physics_ , 160(2):720–742, 2000.
  * Kurganov et al. [2001] Alexander Kurganov, Sebastian Noelle, and Guergana Petrova.  Semidiscrete central-upwind schemes for hyperbolic conservation laws and hamilton–jacobi equations.  _SIAM Journal on Scientific Computing_ , 23(3):707–740, 2001.
  * Greenshields et al. [2010] Christopher J Greenshields, Henry G Weller, Luca Gasparini, and Jason M Reese.  Implementation of semi-discrete, non-staggered central schemes in a colocated, polyhedral, finite volume framework, for high-speed viscous flows.  _International Journal for Numerical Methods in Fluids_ , 63(1):1–21, 2010.
  * Marcantoni et al. [2012] Luis F Gutiérrez Marcantoni, José P Tamagno, and Sergio A Elaskar.  High speed flow simulation using openfoam.  _Mecánica Computacional_ , 31(16):2939–2959, 2012.
  * Mouton [2007] Christopher Andre Mouton.  _Transition between regular reflection and Mach reflection in the dual-solution domain_.  PhD thesis, California Institute of Technology, 2007.

Figure 14: Mach contours at different wedge angles at $\kappa=0.5$. The
transition static case at $\theta=21.5^{\circ}$ is simulated to compare with
the dynamic case. Figure 15: Temperature contours at wedge angle of
$\theta=27^{\circ}$, for the static case and different reduced frequencies,
$\kappa$.


E. Healy: E-mail<EMAIL_ADDRESS>
# Assembly development for the Simons Observatory focal plane readout module

Erin Healy  Department of Physics, Princeton University, Princeton, NJ, USA
Aamir M. Ali  Department of Physics, University of California, Berkeley, CA,
USA  Kam Arnold  Department of Physics, University of California San Diego, La
Jolla, CA, USA  Jason E. Austermann  National Institute of Standards and
Technology, Boulder, CO, USA  James A. Beall  National Institute of Standards
and Technology, Boulder, CO, USA  Sarah Marie Bruno  Department of Physics,
Princeton University, Princeton, NJ, USA  Steve K. Choi  Department of
Physics, Cornell University, Ithaca, NY, USA  Department of Astronomy, Cornell
University, Ithaca, NY, USA  Jake Connors  National Institute of Standards and
Technology, Boulder, CO, USA  Department of Physics, University of Colorado
Boulder, Boulder, CO, USA  Nicholas F. Cothard  Department of Applied and
Engineering Physics, Cornell University, Ithaca, NY, USA  Bradley Dober
National Institute of Standards and Technology, Boulder, CO, USA  Department
of Physics, University of Colorado Boulder, Boulder, CO, USA  Shannon M. Duff
National Institute of Standards and Technology, Boulder, CO, USA  Nicholas
Galitzki  Department of Physics, University of California San Diego, La Jolla,
CA, USA  Gene Hilton  National Institute of Standards and Technology, Boulder,
CO, USA  Shuay-Pwu Patty Ho  Department of Physics, Stanford University, CA,
USA  Johannes Hubmayr  National Institute of Standards and Technology,
Boulder, CO, USA  Bradley R. Johnson  University of Virginia, Department of
Astronomy, Charlottesville, VA, USA  Yaqiong Li  Department of Physics,
Cornell University, Ithaca, NY, USA  Michael J. Link  National Institute of
Standards and Technology, Boulder, CO, USA  Tammy J. Lucas  National Institute
of Standards and Technology, Boulder, CO, USA  Heather McCarrick  Department
of Physics, Princeton University, Princeton, NJ, USA  Michael D. Niemack
Department of Physics, Cornell University, Ithaca, NY, USA  Department of
Astronomy, Cornell University, Ithaca, NY, USA  Maximiliano Silva-Feaver
Department of Physics, University of California San Diego, La Jolla, CA, USA
Rita F. Sonka  Department of Physics, Princeton University, Princeton, NJ, USA
Suzanne Staggs  Department of Physics, Princeton University, Princeton, NJ,
USA  Eve M. Vavagiakis  Department of Physics, Cornell University, Ithaca, NY,
USA  Michael R. Vissers  National Institute of Standards and Technology,
Boulder, CO, USA  Yuhan Wang  Department of Physics, Princeton University,
Princeton, NJ, USA  Benjamin Westbrook  Department of Physics, University of
California, Berkeley, CA, USA  Edward J. Wollack  Department of Applied and
Engineering Physics, Cornell University, Ithaca, NY, USA  Zhilei Xu
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia,
PA, USA  MIT Kavli Institute, Massachusetts Institute of Technology,
Cambridge, MA, USA  Kaiwen Zheng  Department of Physics, Princeton University,
Princeton, NJ, USA

###### Abstract

The Simons Observatory (SO) is a suite of instruments sensitive to temperature
and polarization of the cosmic microwave background (CMB) to be located at
Cerro Toco in the Atacama Desert in Chile. Five telescopes, one large aperture
telescope and four small aperture telescopes, will host roughly 70,000 highly
multiplexed transition edge sensor (TES) detectors operated at 100 mK. Each SO
focal plane module (UFM) couples 1,764 TESes to microwave resonators in a
microwave multiplexing ($\mu$Mux) readout circuit. Before detector
integration, the 100 mK $\mu$Mux components are packaged into multiplexing
modules (UMMs), which are independently validated to ensure they meet SO
performance specifications. Here we present the assembly developments of these
UMM readout packages for mid frequency (90/150 GHz) and ultra high frequency
(220/280 GHz) UFMs.

###### keywords:

CMB instrumentation, microwave multiplexing

##  1 Introduction

###  1.1 Simons Observatory

In the last three decades since the absolute temperature[(1)] and anisotropies
of the CMB were first mapped[(2)], increasingly sophisticated ground-based
instruments, such as BICEP/Keck, SPT, ACT, and POLARBEAR [(3), (4), (5), (6)],
have contributed greatly to our understanding of the universe and its
evolution. The Simons Observatory’s high precision CMB imagers, which will be
located in the Atacama Desert in Chile, will significantly advance the next
generation of cosmological discovery.

To achieve its science goals, SO will deploy a large aperture telescope (LAT)
with a 6 m primary mirror and four 42 cm refracting small aperture telescopes
(SATs) [(7)]. This combination of angular scales will enable SO to measure the
CMB anisotropies from a few degrees to acrminute scale. The LAT will observe
approximately 40% of the sky and the SATs approximately 10%. The SATs are
primarily designed to constrain the tensor-to-scalar ratio _r_ by measuring
large-scale B-mode (divergence-free) polarization. The LAT will measure the
temperature and polarization power spectra of the CMB, which will contribute
to many science targets. These targets include the sum of neutrino masses,
effective number of relativistic species, and other deviations from
$\Lambda$CDM. With a large sky coverage, data from the LAT will be used in
combination with other surveys, including DESI and LSST [(8), (9)], for
further cosmological and astrophysical analysis[(10)].

The LAT will have 13 optics tubes, each of which can contain three detector
arrays, with a total field of view of $\sim$7.8∘ in diameter. SO plans to
populate seven of the thirteen optics tubes. The SATs will each have a single
optics tube that can house up to seven arrays with a rotating cryogenic half-
wave plate [(11)] to modulate the low-frequency atmosphere signal. In total,
SO plans to deploy $\sim$70,000 superconducting transition edge sensor (TES)
detectors [(12), (13)] across the five instruments. Each pixel, which consists
of four TESes, will be sensitive to two orthogonal polarizations and two
frequencies. SO will observe in six frequency bands. The low frequency (LF)
detectors will measure in two bands centered at 30 and 40 GHz. The mid
frequency (MF) bands will be centered at 90 and 150 GHz and the ultra-high
frequency (UHF) detectors at 220 and 280 GHz.

###  1.2 Microwave multiplexing

A key path to improving sensitivity for CMB instruments is increasing the
multiplexing factor, which enables a larger number of detectors to be read out
on a smaller number of wires. Multiplexed readout reduces wiring costs and
complexity, and it is especially important for low-temperature detectors,
where the savings in thermal loading on the focal plane can be significant.
There are several proven multiplexing architectures that amplify the detected
photon signal using superconducting quantum interference devices
(SQUIDs)[(14), (15), (16), (17)]. In recent years, microwave SQUID
multiplexing[(18), (19)] ($\mu$Mux) has emerged as a promising candidate for
unlocking highly densely packed focal planes in CMB instruments[(20)].
Microwave multiplexing involves a large number of superconducting microwave
resonators, each of which encodes a TES’s response at a unique frequency,
fabricated on a single silicon chip. These spectrally non-overlapping signals
share a common transmission line, which is cryogenically amplified and carried
to room temperature electronics with coaxial cables. The high multiplexing
factor targeted by SO is $\mathcal{O}$(1000), larger than any other CMB
instrument to date. Details of the SO multiplexing circuit can be seen in
Figure 1.

---
Figure 1:  Schematic of the $\mu$Mux readout circuit for the Simons
Observatory. Resonators with unique frequencies between 4 and 6 GHz are
capacitively coupled to a common transmission line. Each resonator is
inductively coupled to an rf-SQUID, which is inductively coupled to both a
common flux ramp modulation line and a TES detector channel. The TESes are
biased in series. The dotted gray box encloses all components housed in the SO
focal plane modules. SLAC microresonator radio frequency (SMuRF) electronics
generate the microwave probe tones, as well as the TES bias and flux ramp.
When a photon signal transduces a shift in the resonance of a channel, the RF
signal is amplified at two cryogenic stages (4 K and 40 K) and demodulated
with SMuRF.[(21), (22)]

###  1.3 The universal focal plane modules

The SO detector modules, the universal focal plane modules (UFMs), are
hexagonal packages that house the 100 mK $\mu$Mux components and the detector
array. The package is designed for dense tiling of the detectors. The readout
components are stacked behind the detectors, occupying no focal plane area.
There are three varieties of UFMs, corresponding to the three frequency bands
(LF, MF, UHF). In all three varieties, AlMn TES bolometers[(23)] with
T${}_{c}\sim$160 mK are fabricated on 150 mm silicon wafers. Though the
details of the packaging differ between the three UFM varieties, they share
the same basic design: a vertical stack of detector and $\mu$Mux readout
components that are electrically connected via wire bonds in a package that
can interface to both types of optics tubes (LAT and SAT). In total, SO plans
to deploy 49 UFMs. These proceedings focus on the assembly of the UFM readout
components of the MF and UHF arrays, which are identical.

##  2 Universal multiplexer module design

The 100 mK readout components have their own packaging, called the universal
multiplexing module (UMM), which is mated to the detector array in a UFM, as
described in Section 5. The UMM design detailed here was developed to improve
the RF environment for the microwave resonators as compared to the prototype
described in Ref. 24. In this version, the multiplexer (mux) chips, on which
the resonators and SQUIDs are fabricated, are sandwiched between two grounded
copper plates. The plates provide a nearby RF ground plane to which the mux
chips can be wire bonded. This packaging also creates a box around rows of mux
chips, designed to reduce loss due to box mode coupling to the package, a
standard loss mechanism for microwave resonatorsSage .

###  2.1 Silicon components

At the core of the $\mu$Mux architecture are the multiplexer chips, fabricated
at NIST. Each 4$\times$20 mm2 chip houses 64 readout circuits that can couple
to TESesdober2020microwave . The readout circuit for each channel, as
described in Section 1.2, consists of a microwave resonator inductively
coupled to an rf-SQUID, which is inductively coupled to the flux ramp as well
as the TES channel. The 64 channels are capacitively coupled to a coplanar
waveguide (CPW) transmission line. The UMM is electrically split into two
halves, called “north” and “south”, with two symmetric CPW circuits that share
a common RF ground. Each north/south half of the UMM consists of 14 mux chips
in series for a total of 28 mux chips per UMM.

The routing wafer, fabricated on a 150 mm silicon wafer at NIST, contains the
TES bias circuity (including shunt resistors), the CPW transmission lines, and
flux ramp lines. The twelve bias lines branch into individual channel
circuits, routing from perimeter bond pads (for connection to TES detector
wafer) to corresponding resonator channels on the mux chips. All electrical
lines on the routing wafer are niobium, and they are connected to the detector
wafer and mux chips with 25 $\mu$m diameter aluminum wire bonds. The routing
wafer also has windows for the mux chips and PCBs. On the east side of the
wafer is a cutout window for the microwave launch (RF) PCB; on the west side
is a cutout for the DC PCB, from which the bias lines and flux ramps are
routed.

---
Figure 2:  Left: Exploded rendering of the UMM assembly. Right: Photo of an
assembled UMM without the lid. Twenty-eight multiplexer chips are assembled in
series on a thin copper tray, totaling 1,800 channels. Electrical lines
fabricated on the routing wafer connect the RF and flux ramp signal to the mux
chips via aluminum wire bonds. There are two multiplexed circuits on each UMM,
labeled north and south. The PCBs interface to optics tube readout wiring.

###  2.2 Other components

The mechanical package consists of a thin (500 $\mu$m) copper tray and a
mating lid. All silicon parts and PCBs are mounted on the copper tray. The
tray thickness is constrained by the UFM coupling. Aluminum wire bonds
electrically connect the routing wafer to the detector wafer, so the vertical
distance between these two wafers must not exceed 2 mm to ensure strong wire
bonds.

The tray has extruded features to which the mux chips and the routing wafer
are grounded with wire bonds. The mux chips and routing wafer sit on the thin
tray such that all bonding surfaces are coplanar. This improves conditions for
bonding as well as minimizes the wire bond length, which is critical for
reducing impedance mismatches along the transmission line. Due to the
difference in thermal contractions between silicon and copper, the extruded
features of the copper tray must be machined with tight (25—50 $\mu$m)
tolerances. The copper lid bolts directly to the thin tray along mating
surfaces between mux chip rows. Spring-loaded contact pins glued into the lid
provide clamping force on the routing wafer. There are two PCBs that feed the
electrical lines to the optics tube, one for the RF signal and the other for
the DC (TES bias) and low-frequency signal (flux ramp). A photo of an
assembled UMM without the lid is in Figure 2.

##  3 UMM assembly

The assembly procedure has been developed to ensure minimal degradation of the
multiplexer performance due to assembly. This will enable the UMMs to the meet
requisite performance specifications: a noise level of 65
$\frac{pA}{\sqrt{Hz}}$ and multiplexing factor of 882 (not all resonators are
coupled to detectors). It is critical that the assembly procedures are
reliable and repeatable, as SO plans to field 49 detector modules. The devices
must also be mechanically robust to handling and electrically functional (no
electrical shorts or opens). The major assembly steps include: screening of
components, aligning and clamping the routing wafer to the thin tray, gluing
of multiplexer chips to the thin tray, wire bonding, electrical validation,
and final packaging.

###  3.1 Component screening

All components are visually inspected for defects and checked for mechanical
fit before continuing with assembly. Silicon components undergo additional
electrical screening before they are accepted. The mux chips are individually
electrically validated with a custom probe card at room temperature that
checks for continuity and shorts to ground on the flux ramp and CPW lines. The
resonator performance must also be proven before proceeding with assembly. As
mux chip performance is largely uniform across a fabrication wafer, one sample
mux chip per wafer is packaged for cryogenic testing. The testing checks that
the mux chips meet the design specifications for resonator parameters,
including the internal quality factor (Qi) and yield.

Similarly, the routing wafer undergoes warm and cold screening. Following a
visual inspection for defects, all critical electrical lines (bias lines, CPW
lines, flux ramps) on the wafer are probed for continuity and shorts at room
temperature using a current-controlled probing box. The wafer is then
assembled for cryogenic screening at 100 mK. This involves mounting the wafer
to a PCB and wire bonding the bias lines to the PCB. The wafer is held in
place with BeCu bipod springs bolted to a copper clamp. Each bias line bond
pad on the routing wafer is wire bonded to two pads on the PCB for four-lead
measurements with a Lakeshore 372 resistance bridge. The setup for the routing
wafer cryogenic screening is in Figure 3. Measurements from the cryogenic
testing are used to ensure acceptable shunt resistor values and critical
current limits. Additionally, the bias lines are probed for shorts again at
100 mK, since some shorts detected at 300 K will disappear at low temperatures
(consistent with the apparent shorts being due to conduction through
impurities in silicon).

---
Figure 3:  Photos from a routing wafer screening. Left: The wafer is placed on
a PCB, and the bias lines on the wafer are wire bonded to the PCB for four-
lead measurements. Right: A clamp holds the wafer in place against the PCB and
mounting plate during cryocycle to 100 mK. When cold, the bias line resistance
and critical current of each bias line is measured, as well as resistance
between bias lines.

###  3.2 Assembly alignment and clamping

The next step after screening is to bolt the thin tray onto a copper jig,
which is used for subsequent assembly steps and cold testing. The routing
wafer is then positioned on the thin tray and aligned with two 1.5 mm dowel
pins. The alignment of the silicon parts within the UMM package is critical
for two reasons: (1) limiting offsets between the $\sim$200 $\mu$m wide bond
pads, which would limit automation efficiency and can degrade the strength of
the bonds, and (2) ensuring that the routing wafer is centered on the thin
tray, lest the thermal contractions during cooling to 100 mK break wire bonds
or the wafer itself.

When the routing wafer is properly aligned in the thin tray, it is clamped in
place using o-rings compressed by a washer and screw, which bolts into the
tray. The outer diameter of the compressed o-ring is used as a proxy for
clamping strength. This clamping strategy ensures that the wafer is fully
constrained during wire bonding, which, like alignment, is required for the
automation of wiring bonding and achieving adequate bond strength. A photo and
schematic of the o-ring clamping, as well as a comparison of a properly
compressed o-ring versus an over- or under-compressed o-ring, can be see in
Figure 4.

The mux chips are glued with rubber cement using a Tresky flip-chip bonder,
which has 2 $\mu$m precision. The chips are positioned using alignment marks
in the routing wafer. The use of rubber cement as an adhesive enables the
replacement of mux chips that develop electrical issues, e.g. shorts to
ground. These issues are rare but especially problematic since a short on one
mux chip shorts out the entire multiplexing chain.

The DC PCB, which has bond pads for connection to the routing wafer along one
edge and a connector for a flexible cable on the other side, is bolted to the
thin tray. The RF PCB is mated to the thin tray with silver epoxy for
grounding purposes and bolted for additional mechanical robustness. These
steps are completed before wire bonding for ease of electrical probing during
later assembly steps.

---
Figure 4:  Clamping scheme for UMM assembly. Upper left: A photo of the
routing wafer clamping with o-rings. Upper right: A schematic of the setup.
The routing wafer is clamped at nine locations to ensure minimal movement
during wire bonding. Lower left: Photo of an under-compressed o-ring, in which
the wafer is not clamped at all. Lower center: The o-ring is properly
compressed, providing sufficient clamping force to keep the wafer in place for
wire bonding. Lower right: The o-ring is over-compressed, as it provides no
additional clamping force and is overly stressing the wafer.

###  3.3 UMM wire bonding

There are multiple types of RF and DC wire bonds to the multiplexer chips. Mux
chips have dense (3 per mm) ground bonds around their perimeter directly to
the copper tray and neighboring mux chips. Additional highly densely packed
ground bonds flank the transmission bonds at the CPW launches. The lengths of
these ground bonds and the CPW bonds are minimized to mitigate their impedance
costs while long enough to accommodate relative movement between components
during cryocycling. The RF performance of these bonds in SO multiplexing
devices has been studied and reported on Li, et al.Li . The routing wafer is
also grounded directly to the thin tray at the ends of rows of mux chips. Flux
ramp and TES channel bonds are placed between mux chip and routing wafer,
though their length requirements are less stringent, as they carry low
frequency and DC signals, respectively. The wire bonds for a single mux chip
can be seen in Figure 5.

In total, each UMM has $\sim$8,000 wire bonds. To scale this assembly step for
49 UMMs, all the wire bonding is programmed for automated bonding on a
Delvotec wire bonder. At present, the wire bonding takes roughly 20 hours per
assembly, including time for repairing failed or damaged wires, though we
expect to improve the speed with efficiencies in the programming.

---
Figure 5:  The wire bonds for each multiplexer chip. Dense RF ground bonds to
the copper tray surround each chip. Wire bonds also couple the multiplexer
chip to the transmission line and flux ramp line. Resonator channels are
bonded from the chip to the routing wafer, which routes the channels to the
individual TESes. Left insert: Photo of wire bonds between chip, wafer, and
thin tray. Right insert: Photo of wire bonds between two chips.

###  3.4 Electrical validation

During the wire bonding assembly step, we are also routinely checking for
shorts and opens. Opens may develop when bonds are missing or fail, both of
which can be readily identified with microscope inspection and probing.
Shorts, however, can be much more difficult to track down. For this reason, we
have developed several short checking tools, including a simultaneous short
checking probing box and a program written to triangulate the suspected
location of an electrical short.

Though shorts due to fabrication errors are detected with the initial
screening described in Section 3.1, each detector channel has a trace segment
that is electrically isolated until the wire bonding step. For this reason, we
have developed a short checking protocol employed during wire bonding. This
simultaneous short checking box, which is connected to a current-controlled
probing station, has dials for selecting which combinations of bias lines are
measured. As the automated wire bonding is running, the operator can manually
cycle through combinations to quickly identify any insidious shorts that
develop between bias lines. The setup for simultaneous short checking is shown
in Figure 6. When shorts between bias lines are detected, the wire bonds to
the shorted channel are removed. Assemblies typically have 0—1% of TES
channels left unbonded due to this type of short.

The other important short checking tool, called the universal shorts
diagnostic code (USDC), takes advantage of the high resistivity of niobium at
300 K. Because of the high resistance of the electrical lines, “shorts” due to
wire bonds, or other metal (e.g. wafer traces) in contact, are measured to be
$\mathcal{O}$(100) kOhms. Using the resistance measurements between shorted
bias lines, the location of a short can be estimated by solving a system of
equations. The USDC is also used to triangulate bias line, flux ramp, and CPW
shorts to ground.

---
Figure 6:  Setup for simultaneously probing for electrical shorts between bias
lines while wire bonding. UMM assembly to the left. The box to the right has a
series of dials to select which combination of bias lines to probe. Leads from
this box plug into a current-controlled probing station.

###  3.5 Final packaging

When wire bonding is complete and the integrity of the electrical lines is
validated, the assembly is ready for the final component, the copper lid, to
be added. In the final configuration, six spring-loaded pogo pins, glued with
Stycast 2850 to the lid, provide the clamping force on the routing wafer. Both
the o-rings and dowel pins must be removed to avoid over-constraining the
routing wafer during cryocycle. As an intermediary step, BeCu tabs at the
wafer corners are added to constrain the wafer while the o-rings and pins are
removed. The lid is then bolted to the thin copper tray along surfaces between
rows of mux chips. Due to the proximity to the mux chips, brass screws are
used to avoid magnetic interference. When the lid is fully integrated, the
BeCu corner clamps are removed.

The last step of packaging is to add an interface PCB, which converts the DC
connection from a flexible cable to an MDM connector (compatible with optics
tube wiring). The thin tray remains bolted to the copper jig for testing. A
photo of a fully packaged UMM is in Figure 7.

---
Figure 7:  Photos of a fully packaged UMM. The package is bolted to a copper
jig, used during wire bonding and testing.

##  4 Testing

When UMM assembly is complete, the package is tested in a testbed dilution
refrigerator to confirm it meets mechanical, thermal, and electrical
performance specifications. The testbed readout setup mirrors the SO optics
tube design (Figure 1) with minor modificationsRao . Using SMuRF and a vector
network analyzer (VNA), data are taken to check for transmission, isolation,
noise, and resonator depths and quality factors. Results of the tests have
provided a feedback loop for the assembly development.

##  5 UFM integration

After the performance and robustness of a fully assembled UMM is validated,
the readout package is coupled to a detector array. First, the UMM is unbolted
from the copper jig. It is then transferred to the top of the detector stack
and aligned under microscope. An additional 3,500 wire bonds are placed
between the routing wafer and TES detector wafer to connect TESes to their
resonator channels. Gold bonds are also laid from PdAu pads on the detector
wafer to the gold-plated aluminum feed hornssimon  to create a thermal path
off the detector wafer to the 100 mK bath. The final step of the UFM
integration includes adding a mechanical lid for clamping on the stack-up of
the detectors and UMM components, necessary for good thermal contact as well
as mechanical stability during cool downs and shipping. We have begun testing
prototype UFM assemblies and will report on the details of that assembly when
those procedures have been fully validated.

##  6 Conclusions

The assembly steps outlined here have been developed to meet SO performance
specifications for readout noise and multiplexing factor, as well as
cryomechanical and electrical robustness. Additional assembly steps are
currently under development to shorten production times and further ensure
repeatability between assemblies. One improvement will be validating the thin
trays with a high precision coordinate-measuring machine (CMM) before assembly
integration. We are also working on an upgrade to the simultaneous short
checking setup, which would automate checking for shorts between bias lines
during bonding steps.

To date, we have built two deployment-grade devices, capable of coupling to a
detector array. The design of both UFMs and UMMs will also need additional
attention in a few areas, which could influence assembly steps. For example,
the UMM lid could still be optimized for further reduction of loss due to box
mode coupling by testing different materials and geometries. Another important
study is the magnetic shielding of the multiplexing modules, which is ongoing
Eve .The testbed magnetic shielding differs from the field environment, so
further investigation will be necessary and may result in the addition of
magnetic shielding components. Similarly, the cabling in the testbed
refrigerators allows sufficient strain relieving of the wires near the focal
plane modules, but we plan to develop improved strain relieving directly to
the outer package of the UFM, mitigating any risk of damage during
installation. These improvements will ensure that the fielded UMM devices will
meet the SO performance goals.

###### Acknowledgements.

None of this work would be possible without Bert Harrop and Martina Macakova,
who have been instrumental in developing and executing the assembly techniques
and protocols described here. Zhilei Xu is supported by the Gordon and Betty
Moore Foundation

## References

  * (1) Mather, J. C., Cheng, E., Cottingham, D., Eplee, R., Fixsen, D., Hewagama, T., Isaacman, R., Jesnsen, K., Meyer, S., Noerdlinger, P., Read, S., Rosen, L., Shafer, R., Wright, E., Bennett, C., Boggess, N., Hauser, M., Kelsall, T., Moseley, S. J., Silverberg, R., Smoot, G. F., Weiss, R., and Wilkinson, D., “Measurement of the cosmic microwave background spectrum by the cobe firas instrument,” Astrophysical Journal 420, 439–444 (1994).
  * (2) Smoot, G. F., Bennett, C. L., Kogut, A., Wright, E. L., Aymon, J., Boggess, N. W., Cheng, E. S., de Amici, G., Gulkis, S., Hauser, M. G., Hinshaw, G., Jackson, P. D., Janssen, M., Kaita, E., Kelsall, T., Keegstra, P., Lineweaver, C., Loewenstein, K., Lubin, P., Mather, J., Meyer, S. S., Moseley, S. H., Murdock, T., Rokke, L., Silverberg, R. F., Tenorio, L., Weiss, R., and Wilkinson, D. T., “Structure in the COBE Differential Microwave Radiometer First-Year Maps,” Astrophysical Journal Letters 396, L1 (Sept. 1992).
  * (3) Ade, P., Ahmed, Z., Aikin, R., Alexander, K., Barkats, D., Benton, S., Bischoff, C., Bock, J., Bowens-Rubin, R., Brevik, J., and et al., “Constraints on primordial gravitational waves using planck, wmap, and new bicep2/keck observations through the 2015 season,” Physical Review Letters 121 (Nov 2018).
  * (4) Chown, R., Omori, Y., Aylor, K., Benson, B. A., Bleem, L. E., Carlstrom, J. E., Chang, C. L., Cho, H.-M., Crawford, T. M., Crites, A. T., and et al., “Maps of the southern millimeter-wave sky from combined 2500 deg spt-sz and planck temperature data,” The Astrophysical Journal Supplement Series 239, 10 (Nov 2018).
  * (5) Aiola, S., Calabrese, E., Maurin, L., Naess, S., Schmitt, B. L., Abitbol, M. H., Addison, G. E., Ade, P. A. R., Alonso, D., Amiri, M., Amodeo, S., Angile, E., Austermann, J. E., Baildon, T., Battaglia, N., Beall, J. A., Bean, R., Becker, D. T., Bond, J. R., Bruno, S. M., Calafut, V., Campusano, L. E., Carrero, F., Chesmore, G. E., mei Cho., H., Choi, S. K., Clark, S. E., Cothard, N. F., Crichton, D., Crowley, K. T., Darwish, O., Datta, R., Denison, E. V., Devlin, M. J., Duell, C. J., Duff, S. M., Duivenvoorden, A. J., Dunkley, J., Dünner, R., Essinger-Hileman, T., Fankhanel, M., Ferraro, S., Fox, A. E., Fuzia, B., Gallardo, P. A., Gluscevic, V., Golec, J. E., Grace, E., Gralla, M., Guan, Y., Hall, K., Halpern, M., Han, D., Hargrave, P., Hasselfield, M., Helton, J. M., Henderson, S., Hensley, B., Hill, J. C., Hilton, G. C., Hilton, M., Hincks, A. D., Hložek, R., Ho, S.-P. P., Hubmayr, J., Huffenberger, K. M., Hughes, J. P., Infante, L., Irwin, K., Jackson, R., Klein, J., Knowles, K., Koopman, B., Kosowsky, A., Lakey, V., Li, D., Li, Y., Li, Z., Lokken, M., Louis, T., Lungu, M., MacInnis, A., Madhavacheril, M., Maldonado, F., Mallaby-Kay, M., Marsden, D., McMahon, J., Menanteau, F., Moodley, K., Morton, T., Namikawa, T., Nati, F., Newburgh, L., Nibarger, J. P., Nicola, A., Niemack, M. D., Nolta, M. R., Orlowski-Sherer, J., Page, L. A., Pappas, C. G., Partridge, B., Phakathi, P., Prince, H., Puddu, R., Qu, F. J., Rivera, J., Robertson, N., Rojas, F., Salatino, M., Schaan, E., Schillaci, A., Sehgal, N., Sherwin, B. D., Sierra, C., Sievers, J., Sifon, C., Sikhosana, P., Simon, S., Spergel, D. N., Staggs, S. T., Stevens, J., Storer, E., Sunder, D. D., Switzer, E. R., Thorne, B., Thornton, R., Trac, H., Treu, J., Tucker, C., Vale, L. R., Engelen, A. V., Lanen, J. V., Vavagiakis, E. M., Wagoner, K., Wang, Y., Ward, J. T., Wollack, E. J., Xu, Z., Zago, F., and Zhu, N., “The atacama cosmology telescope: Dr4 maps and cosmological parameters,” (2020).
  * (6) Ade, P. A. R., Aguilar, M., Akiba, Y., Arnold, K., Baccigalupi, C., Barron, D., Beck, D., Bianchini, F., Boettger, D., and et al., “A measurement of the cosmic microwave backgroundb-mode polarization power spectrum at subdegree scales from two years of polarbear data,” The Astrophysical Journal 848, 121 (Oct 2017).
  * (7) Galitzki, N., Baildon, T., Barron, D., Lashner, J., Lee, A. T., Li, Y., Limon, M., Lungu, M., Matsuda, F., Mauskopf, P. D., and et al., “The simons observatory: instrument overview,” Proc. SPIE: Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy IX (Jul 2018).
  * (8) Collaboration, D., Aghamousa, A., Aguilar, J., Ahlen, S., Alam, S., Allen, L. E., Prieto, C. A., Annis, J., Bailey, S., Balland, C., Ballester, O., Baltay, C., Beaufore, L., Bebek, C., Beers, T. C., Bell, E. F., Bernal, J. L., Besuner, R., Beutler, F., Blake, C., Bleuler, H., Blomqvist, M., Blum, R., Bolton, A. S., Briceno, C., Brooks, D., Brownstein, J. R., Buckley-Geer, E., Burden, A., Burtin, E., Busca, N. G., Cahn, R. N., Cai, Y.-C., Cardiel-Sas, L., Carlberg, R. G., Carton, P.-H., Casas, R., Castander, F. J., Cervantes-Cota, J. L., Claybaugh, T. M., Close, M., Coker, C. T., Cole, S., Comparat, J., Cooper, A. P., Cousinou, M. C., Crocce, M., Cuby, J.-G., Cunningham, D. P., Davis, T. M., Dawson, K. S., de la Macorra, A., Vicente, J. D., Delubac, T., Derwent, M., Dey, A., Dhungana, G., Ding, Z., Doel, P., Duan, Y. T., Ealet, A., Edelstein, J., Eftekharzadeh, S., Eisenstein, D. J., Elliott, A., Escoffier, S., Evatt, M., Fagrelius, P., Fan, X., Fanning, K., Farahi, A., Farihi, J., Favole, G., Feng, Y., Fernandez, E., Findlay, J. R., Finkbeiner, D. P., Fitzpatrick, M. J., Flaugher, B., Flender, S., Font-Ribera, A., Forero-Romero, J. E., Fosalba, P., Frenk, C. S., Fumagalli, M., Gaensicke, B. T., Gallo, G., Garcia-Bellido, J., Gaztanaga, E., Fusillo, N. P. G., Gerard, T., Gershkovich, I., Giannantonio, T., Gillet, D., de Rivera, G. G., Gonzalez-Perez, V., Gott, S., Graur, O., Gutierrez, G., Guy, J., Habib, S., Heetderks, H., Heetderks, I., Heitmann, K., Hellwing, W. A., Herrera, D. A., Ho, S., Holland, S., Honscheid, K., Huff, E., Hutchinson, T. A., Huterer, D., Hwang, H. S., Laguna, J. M. I., Ishikawa, Y., Jacobs, D., Jeffrey, N., Jelinsky, P., Jennings, E., Jiang, L., Jimenez, J., Johnson, J., Joyce, R., Jullo, E., Juneau, S., Kama, S., Karcher, A., Karkar, S., Kehoe, R., Kennamer, N., Kent, S., Kilbinger, M., Kim, A. G., Kirkby, D., Kisner, T., Kitanidis, E., Kneib, J.-P., Koposov, S., Kovacs, E., Koyama, K., Kremin, A., Kron, R., Kronig, L., Kueter-Young, A., Lacey, C. G., Lafever, R., Lahav, O., Lambert, A., Lampton, M., Landriau, M., Lang, D., Lauer, T. R., Goff, J.-M. L., Guillou, L. L., Suu, A. L. V., Lee, J. H., Lee, S.-J., Leitner, D., Lesser, M., Levi, M. E., L’Huillier, B., Li, B., Liang, M., Lin, H., Linder, E., Loebman, S. R., Lukić, Z., Ma, J., MacCrann, N., Magneville, C., Makarem, L., Manera, M., Manser, C. J., Marshall, R., Martini, P., Massey, R., Matheson, T., McCauley, J., McDonald, P., McGreer, I. D., Meisner, A., Metcalfe, N., Miller, T. N., Miquel, R., Moustakas, J., Myers, A., Naik, M., Newman, J. A., Nichol, R. C., Nicola, A., da Costa, L. N., Nie, J., Niz, G., Norberg, P., Nord, B., Norman, D., Nugent, P., O’Brien, T., Oh, M., Olsen, K. A. G., Padilla, C., Padmanabhan, H., Padmanabhan, N., Palanque-Delabrouille, N., Palmese, A., Pappalardo, D., Pâris, I., Park, C., Patej, A., Peacock, J. A., Peiris, H. V., Peng, X., Percival, W. J., Perruchot, S., Pieri, M. M., Pogge, R., Pollack, J. E., Poppett, C., Prada, F., Prakash, A., Probst, R. G., Rabinowitz, D., Raichoor, A., Ree, C. H., Refregier, A., Regal, X., Reid, B., Reil, K., Rezaie, M., Rockosi, C. M., Roe, N., Ronayette, S., Roodman, A., Ross, A. J., Ross, N. P., Rossi, G., Rozo, E., Ruhlmann-Kleider, V., Rykoff, E. S., Sabiu, C., Samushia, L., Sanchez, E., Sanchez, J., Schlegel, D. J., Schneider, M., Schubnell, M., Secroun, A., Seljak, U., Seo, H.-J., Serrano, S., Shafieloo, A., Shan, H., Sharples, R., Sholl, M. J., Shourt, W. V., Silber, J. H., Silva, D. R., Sirk, M. M., Slosar, A., Smith, A., Smoot, G. F., Som, D., Song, Y.-S., Sprayberry, D., Staten, R., Stefanik, A., Tarle, G., Tie, S. S., Tinker, J. L., Tojeiro, R., Valdes, F., Valenzuela, O., Valluri, M., Vargas-Magana, M., Verde, L., Walker, A. R., Wang, J., Wang, Y., Weaver, B. A., Weaverdyck, C., Wechsler, R. H., Weinberg, D. H., White, M., Yang, Q., Yeche, C., Zhang, T., Zhao, G.-B., Zheng, Y., Zhou, X., Zhou, Z., Zhu, Y., Zou, H., and Zu, Y., “The desi experiment part i: Science,targeting, and survey design,” (2016).
  * (9) LSST Science Collaboration, Abell, P., Allison, J., Anderson, S., Andrew, J., Angel, J., Armus, L., Arnett, D., Asztalos, S., Axelrod, T., and et al., “LSST Science Book, Version 2.0,” ArXiv e-prints (Dec. 2009).
  * (10) Ade, P., Aguirre, J., Ahmed, Z., Aiola, S., Ali, A., Alonso, D., Alvarez, M. A., Arnold, K., Ashton, P., Austermann, J., and et al., “The simons observatory: science goals and forecasts,” Journal of Cosmology and Astroparticle Physics 2019, 056–056 (Feb 2019).
  * (11) Salatino, M., Lashner, J., Gerbino, M., Simon, S. M., Didier, J., Ali, A., Ashton, P. C., Bryan, S., Chinone, Y., Coughlin, K., Crowley, K. T., Fabbian, G., Galitzki, N., Goeckner-Wald, N., Golec, J. E., Gudmundsson, J. E., Hill, C. A., Keating, B., Kusaka, A., Lee, A. T., McMahon, J., Miller, A. D., Puglisi, G., Reichardt, C. L., Teply, G., Xu, Z., and Zhu, N., “Studies of systematic uncertainties for simons observatory: Polarization modulator related effects,” (2018).
  * (12) Lee, S.-F., Gildemeister, J. M., Holmes, W., Lee, A. T., and Richards, P. L., “Voltage-biased superconducting trainsition-edge bolometer with strong electrothermal with strong electrothermal feedback operated at 370 mk,” Applied Optics 37, 3391–3397 (1998).
  * (13) Irwin, K. D. and Hilton, G. C., “Transition-edge sensors,” Topics in Applied Physics 99, 63–149 (2005).
  * (14) Ahmed, Z., Amiri, M., Benton, S. J., Bock, J. J., Bowens-Rubin, R., Buder, I., Bullock, E., Connors, J., Filippini, J. P., Grayson, J. A., Halpern, M., Hilton, G. C., Hristov, V. V., Hui, H., Irwin, K. D., Kang, J., Karkare, K. S., Karpel, E., Kovac, J. M., Kuo, C. L., Netterfield, C. B., Nguyen, H. T., O’Brient, R., IV, R. W. O., Pryke, C., Reintsema, C. D., Richter, S., Thompson, K. L., Turner, A. D., Vieregg, A. G., Wu, W. L. K., and Yoon, K. W., “Bicep3: A 95 ghz refracting telescope for degree-scale cmb polarization,” Proc. SPIE 9153, Millimeter, Submillimeter, and Far-Infrared Detectors and Instruments for Astronomy VII 91531N (2014).
  * (15) Henderson, S. W., Stevens, J. R., Amiri, M., Austermann, J., Beall, J. A., Chaudhuri, S., Cho, H.-M., Choi, S. K., Cothard, N. F., Crowley, K. T., Duff, S. M., Fitzgerald, C. P., Gallardo, P. A., Halpern, M., Hasselfield, M., Hilton, G., Ho, S.-P. P., Hubmayr, J., Irwin, K. D., Koopman, B. J., Li, D., Li, Y., McMahon, J., Nati, F., Niemack, M., Reintsema, C. D., Salatino, M., Schillaci, A., Schmitt, B. L., Simon, S. M., Staggs, S. T., Vavagiakis, E. M., and Ward, J. T., “Readout of two-kilopixel transition-edge sensor arrays for advanced actpol,” Proc. SPIE 9914, Millimeter, Submillimeter, and Far-Infrared Detectors and Instruments for Astronomy VIII 99141G (2016).
  * (16) Bender, A. N., Ade, P. A. R., Anderson, A. J., Avva, J., Ahmed, Z., Arnold, K., Austermann, J. E., Thakur, R. B., Benson, B. A., Bleem, L. E., Byrum, K., Carlstrom, J. E., Carter, F. W., Chang, C. L., Cho, H. M., Cliche, J. F., Crawford, T. M., Cukierman, A., Czaplewski, D. A., Ding, J., Divan, R., de Haan, T., Dobbs, M. A., Dutcher, D., Everett, W., Gilbert, A., Groh, J. C., Guyser, R., Halverson, N. W., Harke-Hosemann, A., Harrington, N. L., Hattori, K., Henning, J. W., Hilton, G. C., Holzapfel, W. L., Huang, N., Irwin, K. D., Jeong, O., Khaire, T., Korman, M., Kubik, D., Kuo, C. L., Lee, A. T., Leitch, E. M., Lendinez, S., Meyer, S. S., Miller, C. S., Montgomery, J., Nadolski, A., Natoli, T., Nguyen, H., Novosad, V., Padin, S., Pan, Z., Pearson, J., Posada, C. M., Rahlin, A., Reichardt, C. L., Ruhl, J. E., Saliwanchik, B. R., Sayre, J. T., Shariff, J. A., Shirley, I., Shirokoff, E., Smecher, G., Sobrin, J., Stan, L., Stark, A. A., Story, K., Suzuki, A., Tang, Q. Y., Thompson, K. L., Tucker, C., Vanderlinde, K., Vieira, J. D., Wang, G., Whitehorn, N., Yefremenko, V., and Yoon, K. W., “Integrated performance of a frequency domain multiplexing readout in the spt-3g receiver,” Proc. SPIE 9914, Millimeter, Submillimeter, and Far-Infrared Detectors and Instruments for Astronomy VIII 99141D (2016).
  * (17) Suzuki, A., Ade, P., Akiba, Y., Aleman, C., Arnold, K., Baccigalupi, C., Barch, B., Barron, D., Bender, A., Boettger, D., Borrill, J., Chapman, S., Chinone, Y., Cukierman, A., Dobbs, M., Ducout, A., Dunner, R., Elleflot, T., Errard, J., Fabbian, G., Feeney, S., Feng, C., Fujino, T., Fuller, G., Gilbert, A., Goeckner-Wald, N., Groh, J., Haan, T. D., Hall, G., Halverson, N., Hamada, T., Hasegawa, M., Hattori, K., Hazumi, M., Hill, C., Holzapfel, W., Hori, Y., Howe, L., Inoue, Y., Irie, F., Jaehnig, G., Jaffe, A., Jeong, O., Katayama, N., Kaufman, J., Kazemzadeh, K., Keating, B., Kermish, Z., Keskitalo, R., Kisner, T., Kusaka, A., Jeune, M. L., Lee, A., Leon, D., Linder, E., Lowry, L., Matsuda, F., Matsumura, T., Miller, N., Mizukami, K., Montgomery, J., Navaroli, M., Nishino, H., Peloton, J., Poletti, D., Puglisi, G., Rebeiz, G., Raum, C., Reichardt, C., Richards, P., Ross, C., Rotermund, K., Segawa, Y., Sherwin, B., Shirley, I., Siritanasak, P., Stebor, N., Stompor, R., Suzuki, J., Tajima, O., Takada, S., Takakura, S., Takatori, S., Tikhomirov, A., Tomaru, T., Westbrook, B., Whitehorn, N., Yamashita, T., and Zahn, A. Z. . O., “The polarbear-2 and the simons array experiments,” Journal of Low Temperature Physics 184, 3–4 (2016).
  * (18) Irwin, K. D. and Lehnert, K. W., “Microwave squid multiplexer,” Applied Physics Letters 85(11), 2107–2109 (2004).
  * (19) Mates, J. A. B., Hilton, G. C., Irwin, K. D., Vale, L. R., and Lehnert, K. W., “Demonstration of a multiplexer of dissipationless superconducting quantum interference devices,” Applied Physics Letters 92(2), 023514 (2008).
  * (20) Dober, B., Becker, D. T., Bennett, D. A., Bryan, S. A., Duff, S. M., Gard, J. D., Hays-Wehle, J. P., Hilton, G. C., Hubmayr, J., Mates, J. A. B., Reintsema, C. D., Vale, L. R., , and Ullom, J. N., “Microwave squid multiplexer demonstration for cosmic microwave background imagers,” Applied Physics Letters 111, 243510 (2017).
  * (21) Henderson, S. W., Ahmed, Z., Austermann, J., Becker, D., Bennett, D. A., Brown, D., Chaudhuri, S., Cho, H.-M. S., D’Ewart, J. M., Dober, B., Duff, S. M., Dusatko, J. E., Fatigoni, S., Frisch, J. C., Gard, J. D., Halpern, M., Hilton, G. C., Hubmayr, J., Irwin, K. D., Karpel, E. D., Kernasovskiy, S. S., Kuenstner, S. E., Kuo, C.-L., Li, D., Mates, J. A. B., Reintsema, C. D., Smith, S. R., Ullom, J., Vale, L. R., Winkle, D. D. V., Vissers, M., and Yu, C., “Highly-multiplexed microwave squid readout using the slac microresonator radio frequency (smurf) electronics for future cmb and sub-millimeter surveys,” Proc. SPIE 10708, Millimeter, Submillimeter, and Far-Infrared Detectors and Instruments for Astronomy IX 1070819 (2018).
  * (22) Rao, M. S., Silva-Feaver, M., Ali, A., Arnold, K., Ashton, P., Dober, B. J., Duell, C. J., Duff, S. M., Galitzki, N., Healy, E., Henderson, S., Ho, S.-P. P., Hoh, J., Kofman, A. M., Kusaka, A., Lee, A. T., Mangu, A., Mathewson, J., Mauskopf, P., McCarrick, H., Moore, J., Niemack, M. D., Raum, C., Salatino, M., Sasse, T., Seibert, J., Simon, S. M., Staggs, S., Stevens, J. R., Teply, G., Thornton, R., Ullom, J., Vavagiakis, E. M., Westbrook, B., Xu, Z., and Zhu, N., “Simons observatory microwave squid multiplexing readout: Cryogenic rf amplifier and coaxial chain design,” Journal of Low Temperature Physics 199, 807–816 (2020).
  * (23) Deiker, S. W., Doriese, W., Hilton, G. C., Irwin, K. D., Rippard, W. H., Ullom, J. N., Vale, L. R., Ruggiero, S. T., Williams, A., and Young, B. A., “Superconducting transition edge sensor using dilute almn alloys,” Applied Physics Letters 85(11), 2137–2139 (2004).
  * (24) Li, Y., Arnold, K., Atkins, Z., Bruno, S. M., Cothard, N. F., Dober, B., Duell, C. J., Duff, S. M., Gallardo, P. A., Healy, E., Ho, S.-P. P., Hubmayr, J., Keating, B., Lee, A. T., Mangu, A., McCarrick, H., Niemack, M. D., Newburgh, L., Raum, C., Salatino, M., Sasse, T., Silva-Feaver, M., Simon, S. M., Staggs, S., Suzuki, A., Ullom, J., Vavagiakis, E. M., Vissers, M. R., Wang, Y., Westbrook, B., Wollack, E. J., Xu, Z., Zheng, K., and Zhu, N., “Assembly and integration process for the high-density detector array readout modules for the simons observatory,” Journal of Low Temperature Physics 199, 985–993 (2020).
  * (25) Sage, J. M., Bolkhovsky, V., Oliver, W. D., Turek, B., and Welander, P. B., “Study of loss in superconducting coplanar waveguide resonators,” Journal of Applied Physics 109(6), 063915 (2011).
  * (26) Dober, B., Ahmed, Z., Becker, D. T., Bennett, D. A., Connors, J. A., Cukierman, A., D’Ewart, J. M., Duff, S. M., Dusatko, J. E., Frisch, J. C., Gard, J. D., Henderson, S. W., Herbst, R., Hilton, G. C., Hubmayr, J., Mates, J. A. B., Reintsema, C. D., Ruckman, L., Ullom, J. N., Vale, L. R., Winkle, D. D. V., Vasquez, J., Young, E., and Yu, C., “A microwave squid multiplexer optimized for bolometric applications,” (2020).
  * (27) Simon, S. M., Golec, J. E., Ali, A., Austermann, J., Beall, J. A., Bruno, S. M. M., Choi, S. K., Crowley, K. T., Dicker, S., Dober, B., and et al., “Feedhorn development and scalability for simons observatory and beyond,” Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy IX (Jul 2018).
  * (28) Vavagiakis, E. M., Ahmed, Z., Ali, A., Arnold, K., Austermann, J., Bruno, S. M., Choi, S. K., Connors, J., Cothard, N. F., Dicker, S., Dober, B., Duff, S., Fanfani, V., Healy, E., Henderson, S., Ho, S.-P. P., Hoang, D.-T., Hilton, G., Hubmayr, J., Krachmalnicoff, N., Li, Y., Mates, J., McCarrick, H., Nati, F., Niemack, M., Silva-Feaver, M., Staggs, S., Stevens, J., Vissers, M., Ullom, J., Wagoner, K., Xu, Z., and Zhu, N., “The simons observatory: Magnetic sensitivity measurements of microwave squid multiplexers,” (2020).