# Physically Consistent Modeling of Stacked Intelligent Metasurfaces
Implemented with Beyond Diagonal RIS

Matteo Nerini, , Bruno Clerckx M. Nerini and B. Clerckx are with the
Department of Electrical and Electronic Engineering, Imperial College London,
London SW7 2AZ, U.K. (e-mail: {m.nerini20, b.clerckx}@imperial.ac.uk).

###### Abstract

Stacked intelligent metasurface (SIM) has emerged as a technology enabling
wave domain beamforming through multiple stacked reconfigurable intelligent
surfaces (RISs). SIM has been implemented so far with diagonal RIS (D-RIS),
while SIM implemented with beyond diagonal RIS (BD-RIS) remains unexplored.
Furthermore, a model of SIM accounting for mutual coupling is not yet
available. To fill these gaps, we derive a physically consistent channel model
for SIM-aided systems and clarify the assumptions needed to obtain the
simplified model used in related works. Using this model, we show that 1-layer
SIM implemented with BD-RIS achieves the performance upper bound with limited
complexity.

###### Index Terms:

Beyond diagonal reconfigurable intelligent surface (BD-RIS), stacked
intelligent metasurface (SIM).

##  I Introduction

Recently, a novel transceiver design based on stacked intelligent metasurfaces
(SIMs) has been proposed [1, 2]. A SIM consists of multiple stacked layers,
each being a reconfigurable intelligent surface (RIS) allowing the
transmission of the incident electromagnetic (EM) wave in a controllable
manner. Given the presence of multiple reconfigurable layers, a SIM can
efficiently perform beamforming in the EM wave domain [1] and implement
holographic multiple-input multiple-output (MIMO) communications [2].
Specifically, SIM technology is characterized by three main benefits that make
it appealing for future wireless communications. First, SIMs can replace
conventional digital beamforming and remove the necessity for high-resolution
digital-to-analog converter (DAC) and analog-to-digital converter (ADC),
reducing the hardware cost. Second, SIM can reduce the number of needed radio-
frequency (RF) chains, consequently decreasing energy consumption. Third, SIM
can remove the latency due to the precoding processing since it is performed
in the EM wave domain.

In previous works, SIM layers have been implemented using diagonal RIS
(D-RIS), which is the conventional RIS architecture characterized by a
diagonal phase shift matrix [1, 2]. However, to overcome the limited
flexibility of D-RIS, beyond diagonal RIS (BD-RIS) has been recently proposed
as a generalization of D-RIS [3]. The novelty introduced in BD-RIS is the
presence of tunable impedance components interconnecting the RIS elements to
each other, enabling the design of RIS with a scattering matrix that is not
limited to being diagonal. Depending on the location of these tunable
interconnections, multiple BD-RIS architectures have been proposed, such as
fully-/group-connected RISs [4] and tree-/forest-connected RISs [5]. Given the
promising performance enabled by BD-RIS, in this study, we model and compare
SIMs implemented with D-RIS and BD-RIS.

To thoroughly compare SIMs built through D-RIS and BD-RIS, a physically
consistent model of SIM-aided communication systems is needed. Despite a path-
loss model for SIM-aided systems has been proposed in [6], a physically
consistent model of SIM accounting for the EM mutual coupling effects is not
yet available. Nevertheless, EM-compliant analyses of RIS-aided systems have
been developed through multiport network theory using the scattering [4],
impedance [7], and admittance [8] parameters. Furthermore, the relationship
between these three analyses has been investigated in [8, 9]. Since multiport
network theory has successfully produced physically consistent models for RIS-
aided systems, in this study, we employ this theory to model SIM-aided
systems.

The contributions of this study are summarized as follows. First, we model a
SIM-aided communication system through multiport network theory and derive a
general channel model, accounting for the mutual coupling effects at the
transmitter, SIM layers, and receiver. Second, we simplify the derived model
into a simplified channel model, consistent with the model widely used in
recent literature on SIM, clarifying for the first time which are the required
assumptions. Third, we analyze and compare SIM architectures implemented with
D-RIS and BD-RIS. Our theoretical analysis corroborated by numerical results
shows that 1-layer SIM implemented with BD-RIS achieves the performance upper
bound with limited circuit complexity, and that any $L$-layer SIM implemented
with D-RIS is suboptimal.

Figure 1: SIM-aided communication system diagram.

##  II SIM-Aided System Model Based on
Multiport Network Theory

Consider a SIM-aided communication system between an $M$-antenna transmitter
and a $K$-antenna receiver, as represented in Fig. 1. The SIM is deployed at
the transmitter and is composed of $L$ layers, each being an $N$-element RIS.
We model this SIM-aided system through multiport network theory as follows
[10, Chapter 4].

At the transmitter, the $m$th antenna is connected in series with a source
voltage $v_{s,m}$ and a source impedance $Z_{T,m}$, for $m=1,\ldots,M$.
Defining $\mathbf{a}_{T}\in\mathbb{C}^{M\times 1}$ and
$\mathbf{b}_{T}\in\mathbb{C}^{M\times 1}$ as the reflected and incident waves
at the transmitter, we have

$\mathbf{a}_{T}=\mathbf{b}_{s,T}+\mathbf{\Gamma}_{T}\mathbf{b}_{T},$ (1)

where $\mathbf{\Gamma}_{T}\in\mathbb{C}^{M\times M}$ is a diagonal matrix
containing the reflection coefficients of the source impedances, i.e.,

$\mathbf{\Gamma}_{T}=\left(\mathbf{Z}_{T}+Z_{0}\mathbf{I}\right)^{-1}\left(\mathbf{Z}_{T}-Z_{0}\mathbf{I}\right),$
(2)

with $\mathbf{Z}_{T}=\textrm{diag}(Z_{T,1},\ldots,Z_{T,M})$ and $Z_{0}$
denoting the reference impedance typically set to $Z_{0}=50$ $\Omega$, and
$\mathbf{b}_{s,T}\in\mathbb{C}^{M\times 1}$ is the source wave vector given by
$\mathbf{b}_{s,T}=(\mathbf{I}-\mathbf{\Gamma}_{T})\mathbf{v}_{s,T}/2$, with
$\mathbf{v}_{s,T}=[v_{s,1},\ldots,v_{s,M}]^{T}$ [8].

The SIM is a cascade of $L$ blocks, each consisting of the cascade of a
wireless channel and an RIS. Specifically, the $\ell$th block consists of the
cascade of the wireless channel between the $(\ell-1)$th and the $\ell$th SIM
layer and the RIS implementing the $\ell$th SIM layer, for $\ell=2,\ldots,L$.
We model the channel between the $(\ell-1)$th and the $\ell$th SIM layer as a
$2N$-port network with scattering matrix
$\mathbf{H}^{(\ell)}\in\mathbb{C}^{2N\times 2N}$ given by

$\mathbf{H}^{(\ell)}=\begin{bmatrix}\mathbf{H}_{11}^{(\ell)}&\mathbf{H}_{12}^{(\ell)}\\\
\mathbf{H}_{21}^{(\ell)}&\mathbf{H}_{22}^{(\ell)}\end{bmatrix},$ (3)

where $\mathbf{H}_{11}^{(\ell)}\in\mathbb{C}^{N\times N}$ and
$\mathbf{H}_{22}^{(\ell)}\in\mathbb{C}^{N\times N}$ refer to the antenna
mutual coupling at the $(\ell-1)$th and the $\ell$th SIM layer, respectively,
and $\mathbf{H}_{21}^{(\ell)}\in\mathbb{C}^{N\times N}$ refer to the
transmission scattering matrix from the $(\ell-1)$th to the $\ell$th SIM
layer, for $\ell=2,\ldots,L$. Beside, we model the RIS at the $\ell$th SIM
layer through its scattering matrix
$\boldsymbol{\Theta}^{(\ell)}\in\mathbb{C}^{2N\times 2N}$ given by

$\boldsymbol{\Theta}^{(\ell)}=\begin{bmatrix}\boldsymbol{\Theta}_{11}^{(\ell)}&\boldsymbol{\Theta}_{12}^{(\ell)}\\\
\boldsymbol{\Theta}_{21}^{(\ell)}&\boldsymbol{\Theta}_{22}^{(\ell)}\end{bmatrix},$
(4)

for $\ell=2,\ldots,L$, which is symmetric in the case of a reciprocal RIS and
whose structure depends on the RIS architecture. In the case the SIM layers
are implemented through conventional D-RISs, we have
$\boldsymbol{\Theta}_{11}^{(\ell)}=\mathbf{0}$ and
$\boldsymbol{\Theta}_{22}^{(\ell)}=\mathbf{0}$ since the RISs work in
transmissive mode [11], and

$\boldsymbol{\Theta}_{12}^{(\ell)}=\boldsymbol{\Theta}_{21}^{(\ell)}=\text{diag}\left(e^{j\theta_{1}^{(\ell)}},\ldots,e^{j\theta_{N}^{(\ell)}}\right),$
(5)

assuming the RISs to be lossless [1, 2]. However, these constraints valid for
D-RISs can be relaxed into the more general constraint
$\boldsymbol{\Theta}^{(\ell)H}\boldsymbol{\Theta}^{(\ell)}=\mathbf{I}$ by
implementing the SIM layers through BD-RISs [11].

Note that the first block, i.e., $\ell=1$, consists of the cascade of the
wireless channel between the transmitter and the first SIM layer and the RIS
implementing the first SIM layer. Thus, the scattering matrix of the wireless
channel of the first block $\mathbf{H}^{(1)}\in\mathbb{C}^{(M+N)\times(M+N)}$
is given as in (3), where $\mathbf{H}_{11}^{(1)}\in\mathbb{C}^{M\times M}$
refer to the antenna mutual coupling at the transmitter. Besides the
scattering matrix of the first SIM layer $\boldsymbol{\Theta}^{(1)}$ is given
as in (4).

The wireless channel between the $L$th SIM layer and the receiver is modeled
as an $(N+K)$-port network with scattering matrix
$\mathbf{H}^{R}\in\mathbb{C}^{(N+K)\times(N+K)}$ given by

$\mathbf{H}^{R}=\begin{bmatrix}\mathbf{H}_{11}^{R}&\mathbf{H}_{12}^{R}\\\
\mathbf{H}_{21}^{R}&\mathbf{H}_{22}^{R}\end{bmatrix},$ (6)

where $\mathbf{H}_{11}^{R}\in\mathbb{C}^{N\times N}$ and
$\mathbf{H}_{22}^{R}\in\mathbb{C}^{K\times K}$ refer to the antenna mutual
coupling at the $L$th SIM layer and the receiver, respectively, and
$\mathbf{H}_{21}^{R}\in\mathbb{C}^{K\times N}$ refer to the transmission
scattering matrix from the $L$th SIM layer to the receiver.

At the receiver, the $k$th antenna is connected in series with a load
impedance $Z_{R,k}$, for $k=1,\ldots,K$. Defining
$\mathbf{a}_{R}\in\mathbb{C}^{K\times 1}$ and
$\mathbf{b}_{R}\in\mathbb{C}^{K\times 1}$ the reflected and incident waves at
the receiver, we have

$\mathbf{a}_{R}=\mathbf{\Gamma}_{R}\mathbf{b}_{R},$ (7)

where $\mathbf{\Gamma}_{R}\in\mathbb{C}^{K\times K}$ is a diagonal matrix
containing the reflection coefficients of the load impedances, i.e.,

$\mathbf{\Gamma}_{R}=\left(\mathbf{Z}_{R}+Z_{0}\mathbf{I}\right)^{-1}\left(\mathbf{Z}_{R}-Z_{0}\mathbf{I}\right),$
(8)

with $\mathbf{Z}_{R}=\textrm{diag}(Z_{R,1},\ldots,Z_{R,K})$ [8].

##  III General SIM-Aided Channel Model

Our goal is to determine the expression of the channel matrix
$\mathbf{H}\in\mathbb{C}^{K\times M}$ relating the voltage vector at the
transmitter $\mathbf{v}_{T}\in\mathbb{C}^{M\times 1}$ and the voltage vector
at the receiver $\mathbf{v}_{R}\in\mathbb{C}^{K\times 1}$ though
$\mathbf{v}_{R}=\mathbf{H}\mathbf{v}_{T}$, where $\mathbf{v}_{T}$ and
$\mathbf{v}_{R}$ are given by

$\mathbf{v}_{T}=\mathbf{a}_{T}+\mathbf{b}_{T},\>\mathbf{v}_{R}=\mathbf{a}_{R}+\mathbf{b}_{R},$
(9)

according to multiport network theory [10, Chapter 4]. To this end, we
introduce a proposition enabling the analysis of the considered SIM-aided
system.

######  Proposition 1.

Consider the cascade system represented in Fig. 2, consisting of an
$N_{P}$-port network, with scattering matrix
$\mathbf{P}\in\mathbb{C}^{N_{P}\times N_{P}}$ given by

$\mathbf{P}=\begin{bmatrix}\mathbf{P}_{11}&\mathbf{P}_{12}\\\
\mathbf{P}_{21}&\mathbf{P}_{22}\end{bmatrix},$ (10)

where $\mathbf{P}_{11}\in\mathbb{C}^{N_{1}\times N_{1}}$ and
$\mathbf{P}_{22}\in\mathbb{C}^{N_{2}\times N_{2}}$, with $N_{P}=N_{1}+N_{2}$,
and an $N_{Q}$-port network, with scattering matrix
$\mathbf{Q}\in\mathbb{C}^{N_{Q}\times N_{Q}}$ given by

$\mathbf{Q}=\begin{bmatrix}\mathbf{Q}_{11}&\mathbf{Q}_{12}\\\
\mathbf{Q}_{21}&\mathbf{Q}_{22}\end{bmatrix},$ (11)

where $\mathbf{Q}_{11}\in\mathbb{C}^{N_{2}\times N_{2}}$ and
$\mathbf{Q}_{22}\in\mathbb{C}^{N_{3}\times N_{3}}$, with $N_{Q}=N_{2}+N_{3}$.
In this cascade, the last $N_{2}$ ports of the $N_{P}$-port network are
individually connected to the first $N_{2}$ ports of the $N_{Q}$-port network.
Thus, the whole cascade can be regarded as an $N_{R}$-port network, where
$N_{R}=N_{1}+N_{3}$, with scattering matrix
$\mathbf{R}\in\mathbb{C}^{N_{R}\times N_{R}}$ given by

$\mathbf{R}=\begin{bmatrix}\mathbf{R}_{11}&\mathbf{R}_{12}\\\
\mathbf{R}_{21}&\mathbf{R}_{22}\end{bmatrix},$ (12)

with

$\mathbf{R}_{11}=\mathbf{P}_{11}+\mathbf{P}_{12}\left(\mathbf{I}-\mathbf{Q}_{11}\mathbf{P}_{22}\right)^{-1}\mathbf{Q}_{11}\mathbf{P}_{21},$
(13)
$\mathbf{R}_{12}=\mathbf{P}_{12}\left(\mathbf{I}-\mathbf{Q}_{11}\mathbf{P}_{22}\right)^{-1}\mathbf{Q}_{12},$
(14)
$\mathbf{R}_{21}=\mathbf{Q}_{21}\left(\mathbf{I}-\mathbf{P}_{22}\mathbf{Q}_{11}\right)^{-1}\mathbf{P}_{21},$
(15)
$\mathbf{R}_{22}=\mathbf{Q}_{22}+\mathbf{Q}_{21}\left(\mathbf{I}-\mathbf{P}_{22}\mathbf{Q}_{11}\right)^{-1}\mathbf{P}_{22}\mathbf{Q}_{12}.$
(16)

###### Proof.

Please refer to the Appendix. ∎

Figure 2: Cascade of two multiport networks.

To derive the expression of $\mathbf{H}$, we first recall that the $\ell$th
block of the SIM is formed by the cascade of the channel $\mathbf{H}^{(\ell)}$
and the RIS $\boldsymbol{\Theta}^{(\ell)}$, given by (3) and (4),
respectively. Thus, the scattering matrix of the $\ell$th block is given by

$\mathbf{S}^{(\ell)}=\begin{bmatrix}\mathbf{S}_{11}^{(\ell)}&\mathbf{S}_{12}^{(\ell)}\\\
\mathbf{S}_{21}^{(\ell)}&\mathbf{S}_{22}^{(\ell)}\end{bmatrix},$ (17)

with

$\mathbf{S}_{11}^{(\ell)}=\mathbf{H}_{11}^{(\ell)}+\mathbf{H}_{12}^{(\ell)}\left(\mathbf{I}-\boldsymbol{\Theta}_{11}^{(\ell)}\mathbf{H}_{22}^{(\ell)}\right)^{-1}\boldsymbol{\Theta}_{11}^{(\ell)}\mathbf{H}_{21}^{(\ell)},$
(18)
$\mathbf{S}_{12}^{(\ell)}=\mathbf{H}_{12}^{(\ell)}\left(\mathbf{I}-\boldsymbol{\Theta}_{11}^{(\ell)}\mathbf{H}_{22}^{(\ell)}\right)^{-1}\boldsymbol{\Theta}_{12}^{(\ell)},$
(19)
$\mathbf{S}_{21}^{(\ell)}=\boldsymbol{\Theta}_{21}^{(\ell)}\left(\mathbf{I}-\mathbf{H}_{22}^{(\ell)}\boldsymbol{\Theta}_{11}^{(\ell)}\right)^{-1}\mathbf{H}_{21}^{(\ell)},$
(20)
$\mathbf{S}_{22}^{(\ell)}=\boldsymbol{\Theta}_{22}^{(\ell)}+\boldsymbol{\Theta}_{21}^{(\ell)}\left(\mathbf{I}-\mathbf{H}_{22}^{(\ell)}\boldsymbol{\Theta}_{11}^{(\ell)}\right)^{-1}\mathbf{H}_{22}^{(\ell)}\boldsymbol{\Theta}_{12}^{(\ell)},$
(21)

for $\ell=1,\ldots,L$, following Proposition 16. Given $\mathbf{S}^{(\ell)}$,
the equivalent scattering matrix including the first $\ell$ blocks of the SIM
can be determined recursively by

$\bar{\mathbf{S}}^{(\ell)}=\begin{bmatrix}\bar{\mathbf{S}}_{11}^{(\ell)}&\bar{\mathbf{S}}_{12}^{(\ell)}\\\
\bar{\mathbf{S}}_{21}^{(\ell)}&\bar{\mathbf{S}}_{22}^{(\ell)}\end{bmatrix},$
(22)

with

$\bar{\mathbf{S}}_{11}^{(\ell)}=\bar{\mathbf{S}}_{11}^{(\ell-1)}+\bar{\mathbf{S}}_{12}^{(\ell-1)}\left(\mathbf{I}-\mathbf{S}_{11}^{(\ell)}\bar{\mathbf{S}}_{22}^{(\ell-1)}\right)^{-1}\mathbf{S}_{11}^{(\ell)}\bar{\mathbf{S}}_{21}^{(\ell-1)},$
(23)
$\bar{\mathbf{S}}_{12}^{(\ell)}=\bar{\mathbf{S}}_{12}^{(\ell-1)}\left(\mathbf{I}-\mathbf{S}_{11}^{(\ell)}\bar{\mathbf{S}}_{22}^{(\ell-1)}\right)^{-1}\mathbf{S}_{12}^{(\ell)},$
(24)
$\bar{\mathbf{S}}_{21}^{(\ell)}=\mathbf{S}_{21}^{(\ell)}\left(\mathbf{I}-\bar{\mathbf{S}}_{22}^{(\ell-1)}\mathbf{S}_{11}^{(\ell)}\right)^{-1}\bar{\mathbf{S}}_{21}^{(\ell-1)},$
(25)
$\bar{\mathbf{S}}_{22}^{(\ell)}=\mathbf{S}_{22}^{(\ell)}+\mathbf{S}_{21}^{(\ell)}\left(\mathbf{I}-\bar{\mathbf{S}}_{22}^{(\ell-1)}\mathbf{S}_{11}^{(\ell)}\right)^{-1}\bar{\mathbf{S}}_{22}^{(\ell-1)}\mathbf{S}_{12}^{(\ell)},$
(26)

as a function of $\bar{\mathbf{S}}^{(\ell-1)}$ and $\mathbf{S}^{(\ell)}$,
where $\bar{\mathbf{S}}^{(1)}=\mathbf{S}^{(1)}$, according to Proposition 16.
In particular, $\bar{\mathbf{S}}^{(L)}$ represents the equivalent scattering
matrix including all the $L$ blocks of the SIM. From $\bar{\mathbf{S}}^{(L)}$
and the channel $\mathbf{H}^{R}$, the scattering matrix of the entire channel
between the transmitter and the receiver is given by the cascade of
$\bar{\mathbf{S}}^{(L)}$ and $\mathbf{H}^{R}$, and writes as

$\mathbf{S}=\begin{bmatrix}\mathbf{S}_{11}&\mathbf{S}_{12}\\\
\mathbf{S}_{21}&\mathbf{S}_{22}\end{bmatrix},$ (27)

with

$\mathbf{S}_{11}=\bar{\mathbf{S}}_{11}^{(L)}+\bar{\mathbf{S}}_{12}^{(L)}\left(\mathbf{I}-\mathbf{H}_{11}^{R}\bar{\mathbf{S}}_{22}^{(L)}\right)^{-1}\mathbf{H}_{11}^{R}\bar{\mathbf{S}}_{21}^{(L)},$
(28)
$\mathbf{S}_{12}=\bar{\mathbf{S}}_{12}^{(L)}\left(\mathbf{I}-\mathbf{H}_{11}^{R}\bar{\mathbf{S}}_{22}^{(L)}\right)^{-1}\mathbf{H}_{12}^{R},$
(29)
$\mathbf{S}_{21}=\mathbf{H}_{21}^{R}\left(\mathbf{I}-\bar{\mathbf{S}}_{22}^{(L)}\mathbf{H}_{11}^{R}\right)^{-1}\bar{\mathbf{S}}_{21}^{(L)},$
(30)
$\mathbf{S}_{22}=\mathbf{H}_{22}^{R}+\mathbf{H}_{21}^{R}\left(\mathbf{I}-\bar{\mathbf{S}}_{22}^{(L)}\mathbf{H}_{11}^{R}\right)^{-1}\bar{\mathbf{S}}_{22}^{(L)}\mathbf{H}_{12}^{R},$
(31)

according to Proposition 16.

To express $\mathbf{H}$ as a function of $\mathbf{S}$, we consider
$\mathbf{Z}_{T}=Z_{0}\mathbf{I}$ and $\mathbf{Z}_{R}=Z_{0}\mathbf{I}$ for
simplicity, so that $\mathbf{\Gamma}_{T}=\mathbf{0}$ and
$\mathbf{\Gamma}_{R}=\mathbf{0}$ according to (2) and (8), respectively.
Consequently, $\mathbf{H}$ can be derived by solving the system

$\displaystyle\begin{bmatrix}\mathbf{b}_{T}\\\
\mathbf{b}_{R}\end{bmatrix}=\begin{bmatrix}\mathbf{S}_{11}&\mathbf{S}_{12}\\\
\mathbf{S}_{21}&\mathbf{S}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{T}\\\
\mathbf{a}_{R}\end{bmatrix},$ (32)
$\displaystyle\mathbf{a}_{T}=\mathbf{b}_{s,T},\>\mathbf{a}_{R}=\mathbf{0},$
(33) $\displaystyle\eqref{eq:vtvr},$ (34)

where (32) follows the definition of scattering matrix [10, Chapter 4] and
(33) is given by (1) and (7). Since it is possible to show that system
(32)-(34) yields
$\mathbf{v}_{R}=\mathbf{S}_{21}\left(\mathbf{I}+\mathbf{S}_{11}\right)^{-1}\mathbf{v}_{T}$,
we have

$\mathbf{H}=\mathbf{S}_{21}\left(\mathbf{I}+\mathbf{S}_{11}\right)^{-1},$ (35)

giving the general expression of a SIM-aided channel. Remarkably, it is hard
to understand the role of the reconfigurable scattering matrices
$\boldsymbol{\Theta}^{(\ell)}$ in (35). For this reason, we simplify this
channel model and show under which assumptions it boils down to the easier
model widely used in recent works on SIM.

##  IV Simplified SIM-Aided Channel Model

To simplify the channel model in (35), we assume that the unilateral
approximation is valid in all the channels $\mathbf{H}^{(\ell)}$, i.e.,
$\mathbf{H}_{12}^{(\ell)}=\mathbf{0}$, for $\ell=1,\ldots,L$ [12]. In other
words, we assume that the electrical properties at the $(\ell-1)$th SIM layer
are independent of the electrical properties at the $\ell$th SIM layer (where
the $0$th SIM layer means the transmitter). Given the short transmission
distances in a SIM architecture, extensive electromagnetic simulations are
needed to verify the validity of this assumption, which is beyond the scope of
this study. In addition, we assume no mutual coupling between the transmit
antennas and between the RIS elements, i.e.,
$\mathbf{H}_{11}^{(\ell)}=\mathbf{0}$ and
$\mathbf{H}_{22}^{(\ell)}=\mathbf{0}$, for $\ell=1,\ldots,L$, and
$\mathbf{H}_{11}^{R}=\mathbf{0}$. With these assumptions, the scattering
matrix of the $\ell$th block of the SIM given by (17) simplifies as

$\mathbf{S}^{(\ell)}=\begin{bmatrix}\mathbf{0}&\mathbf{0}\\\
\boldsymbol{\Theta}_{21}^{(\ell)}\mathbf{H}_{21}^{(\ell)}&\boldsymbol{\Theta}_{22}^{(\ell)}\end{bmatrix},$
(36)

for $\ell=1,\ldots,L$. Thus, the equivalent scattering matrix of the first
$\ell$ blocks of the SIM is given recursively as

$\bar{\mathbf{S}}^{(\ell)}=\begin{bmatrix}\bar{\mathbf{S}}_{11}^{(\ell-1)}&\mathbf{0}\\\
\mathbf{S}_{21}^{(\ell)}\bar{\mathbf{S}}_{21}^{(\ell-1)}&\mathbf{S}_{22}^{(\ell)}\end{bmatrix},$
(37)

where $\bar{\mathbf{S}}^{(1)}=\mathbf{S}^{(1)}$. Consequently, the equivalent
scattering matrix including all the $L$ blocks of the SIM can now be written
in closed-form as

$\bar{\mathbf{S}}^{(L)}=\begin{bmatrix}\mathbf{0}&\mathbf{0}\\\
\boldsymbol{\Theta}_{21}^{(L)}\mathbf{H}_{21}^{(L)}\cdots\boldsymbol{\Theta}_{21}^{(2)}\mathbf{H}_{21}^{(2)}\boldsymbol{\Theta}_{21}^{(1)}\mathbf{H}_{21}^{(1)}&\boldsymbol{\Theta}_{22}^{(L)}\end{bmatrix}.$
(38)

Given $\bar{\mathbf{S}}^{(L)}$ and $\mathbf{H}^{R}$, the scattering matrix of
the entire channel between the transmitter and the receiver writes as

$\mathbf{S}=\begin{bmatrix}\mathbf{0}&\mathbf{0}\\\
\mathbf{H}_{21}^{R}\boldsymbol{\Theta}_{21}^{(L)}\mathbf{H}_{21}^{(L)}\cdots\boldsymbol{\Theta}_{21}^{(1)}\mathbf{H}_{21}^{(1)}&\mathbf{H}_{22}^{R}+\mathbf{H}_{21}^{R}\boldsymbol{\Theta}_{22}^{(L)}\mathbf{H}_{12}^{R}\end{bmatrix},$
(39)

by Proposition 16. By substituting (39) into (35), and using the auxiliary
notation $\bar{\mathbf{H}}^{R}=\mathbf{H}_{21}^{R}$,
$\bar{\boldsymbol{\Theta}}^{(\ell)}=\boldsymbol{\Theta}_{21}^{(\ell)}$, and
$\bar{\mathbf{H}}^{(\ell)}=\mathbf{H}_{21}^{(\ell)}$, for $\ell=1,\ldots,L$,
we finally obtain

$\mathbf{H}=\bar{\mathbf{H}}^{R}\bar{\boldsymbol{\Theta}}^{(L)}\bar{\mathbf{H}}^{(L)}\cdots\bar{\boldsymbol{\Theta}}^{(2)}\bar{\mathbf{H}}^{(2)}\bar{\boldsymbol{\Theta}}^{(1)}\bar{\mathbf{H}}^{(1)},$
(40)

which is the simplified channel model, in agreement with the model widely
utilized in recent works [1, 2, 6].

##  V Implementing SIM with D-RIS and BD-RIS

Consider a single-input single-output (SISO) system, i.e., $M=1$ and $K=1$,
where we maximize the channel gain

$\left|h\right|^{2}=\left|\bar{\mathbf{h}}^{R}\bar{\boldsymbol{\Theta}}^{(L)}\bar{\mathbf{H}}^{(L)}\cdots\bar{\boldsymbol{\Theta}}^{(2)}\bar{\mathbf{H}}^{(2)}\bar{\boldsymbol{\Theta}}^{(1)}\bar{\mathbf{h}}^{(1)}\right|^{2},$
(41)

by optimizing the SIM, i.e., $\bar{\boldsymbol{\Theta}}^{(\ell)}$, for
$\ell=1,\ldots,L$.

In the case of a SIM implemented with D-RIS,
$\bar{\boldsymbol{\Theta}}^{(\ell)}$ are subject to (5) and a generous upper
bound on the achievable channel gain is given by

$\left|h^{\text{D-RIS}}\right|^{2}\leq\left\|\bar{\mathbf{h}}^{R}\right\|^{2}\left\|\bar{\mathbf{H}}^{(L)}\right\|^{2}\cdots\left\|\bar{\mathbf{H}}^{(2)}\right\|^{2}\left\|\bar{\mathbf{h}}^{(1)}\right\|^{2},$
(42)

following the submultiplicativity property of the spectral norm and noticing
that $\|\bar{\boldsymbol{\Theta}}^{(\ell)}\|=1$, for $\ell=1,\ldots,L$. To
optimize the SIM, we propose an iterative algorithm that optimizes a matrix
$\bar{\boldsymbol{\Theta}}^{(\ell)}$ at each iteration through a closed-form
globally optimal solution. Specifically, at each iteration,
$\bar{\boldsymbol{\Theta}}^{(\ell)}$ is updated by introducing
$\mathbf{u}^{(\ell)}=\bar{\mathbf{h}}^{R}\bar{\boldsymbol{\Theta}}^{(L)}\bar{\mathbf{H}}^{(L)}\cdots\bar{\boldsymbol{\Theta}}^{(\ell+1)}\bar{\mathbf{H}}^{(\ell+1)}$
and
$\mathbf{v}^{(\ell)}=\bar{\mathbf{H}}^{(\ell)}\bar{\boldsymbol{\Theta}}^{(\ell-1)}\bar{\mathbf{H}}^{(\ell-1)}\cdots\bar{\boldsymbol{\Theta}}^{(1)}\bar{\mathbf{h}}^{(1)}$,
and setting
$\theta_{n}^{(\ell)}=-\mathrm{arg}([\mathbf{u}^{(\ell)}]_{n}[\mathbf{v}^{(\ell)}]_{n})$,
for $n=1,\ldots,N$. The algorithm iterates until convergence of the objective
(41). In terms of circuit complexity, when the SIM layers are implemented
through D-RISs working in transmissive mode, the number of tunable impedances
in the circuit topology is $3N$ per layer [11], yielding $3NL$ total tunable
impedances in the SIM architecture.

In the case of a SIM implemented with BD-RIS, we first consider $L=1$ and we
show that this is the optimal number of layers. When $L=1$, (41) boils down to
$|h|^{2}=|\bar{\mathbf{h}}^{R}\bar{\boldsymbol{\Theta}}^{(1)}\bar{\mathbf{h}}^{(1)}|^{2}$,
which can be rewritten as
$|h|^{2}=|[\mathbf{0},\bar{\mathbf{h}}^{R}]\boldsymbol{\Theta}^{(1)}[\bar{\mathbf{h}}^{(1)T},\mathbf{0}]^{T}|^{2}$,
where $\mathbf{0}$ denotes here a $1\times N$ zero vector and
$\boldsymbol{\Theta}^{(1)}$ is subject to
$\boldsymbol{\Theta}^{(1)}=\boldsymbol{\Theta}^{(1)T}$ and
$\boldsymbol{\Theta}^{(1)H}\boldsymbol{\Theta}^{(1)}=\mathbf{I}$.
Interestingly, it has been shown in [13] that $\boldsymbol{\Theta}^{(1)}$ can
be optimized to obtain

$\left|h^{\text{BD-
RIS}}\right|^{2}=\left\|\bar{\mathbf{h}}^{R}\right\|^{2}\left\|\bar{\mathbf{h}}^{(1)}\right\|^{2}.$
(43)

Besides, the performance upper bound in (43) can also be achieved with low-
complexity BD-RIS architectures denoted as tree-connected RISs through a
closed-form solution [5]. Since it holds that
$\|\bar{\mathbf{H}}^{(\ell)}\|<1$ for the wireless channels
$\bar{\mathbf{H}}^{(\ell)}$, the upper bound in (43) is always greater than
(42). Thus, $L=1$ is the optimal number of layers in a SIM implemented with
tree-connected RISs. Furthermore, we have
$\left|h^{\text{D-RIS}}\right|^{2}\leq\left|h^{\text{BD-RIS}}\right|^{2}$ as a
consequence of (42) and (43). Considering a 1-layer SIM implemented through a
tree-connected RIS, the number of needed tunable impedances is $4N-1$ [5].

##  VI Numerical Results

Consider a 3-dimensional coordinate system $(x,y,z)$ in which we locate a SISO
system between a SIM-aided transmitter and a receiver. The transmit antenna is
located at $(0,0,0)$ and the SIM layers are $N_{x}\times N_{y}$ uniform planar
arrays (UPAs) parallel to the $x$-$y$ plane and centered in $(0,0)$. The first
SIM layer is spaced $\lambda$ from the transmit antenna and the SIM layers are
spaced $\lambda$ from each other, where $\lambda$ denotes the wavelength. The
RIS elements in each SIM layer have inter-element distance $\lambda/2$ and we
set $N_{x}=4$ and carrier frequency 28 GHz. The channels
$\bar{\mathbf{H}}^{(\ell)}$, for $\ell=1,\ldots,L$, are modeled using
Rayleigh-Sommerfield’s diffraction theory as

$\left[\bar{\mathbf{H}}^{(\ell)}\right]_{i,j}=\frac{A\cos\left(\alpha_{i,j}^{(\ell)}\right)}{d_{i,j}^{(\ell)}}\left(\frac{1}{2\pi
d_{i,j}^{(\ell)}}-j\frac{1}{\lambda}\right)e^{j2\pi
d_{i,j}^{(\ell)}/\lambda},$ (44)

where $A=(\lambda/2)^{2}$ is the area of each RIS element,
$\alpha_{i,j}^{(\ell)}$ is the angle between the normal of the $(\ell-1)$th
SIM layer and the propagation direction and $d_{i,j}^{(\ell)}$ is the
transmission distance [1]. Besides, the channel $\bar{\mathbf{h}}^{R}$ is
modeled as independent and identically distributed (i.i.d.) Rayleigh
distributed.

In Fig. 3, we report the performance in terms of normalized channel gain,
defined as
$G=|h|^{2}/(\|\bar{\mathbf{h}}^{R}\|^{2}\|\bar{\mathbf{h}}^{(1)}\|^{2})$,
achieved by SIM implemented with D-RIS and BD-RIS. For SIM implemented with
BD-RIS, we consider a 1-layer SIM implemented through a tree-connected RIS,
optimized as proposed in [5]. We make the following observations. First,
1-layer SIMs implemented with BD-RIS always achieve $G=1$, i.e., the
performance upper bound, as expected from (43). Second, SIMs implemented with
D-RIS always underperform 1-layer SIMs implemented with BD-RIS, in agreement
with our theoretical derivations. Third, for a low number of RIS elements $N$,
SIMs with a low number of layers $L$ are beneficial. This is because a low $N$
reduces the norms $\|\bar{\mathbf{H}}^{(\ell)}\|$, for $\ell=1,\ldots,L$,
which are always $\|\bar{\mathbf{H}}^{(\ell)}\|<1$. Consequently, when
$\|\bar{\mathbf{H}}^{(\ell)}\|<<1$, a high $L$ is suboptimal as it limits the
product in the performance upper bound (42). Fourth, for a high value of $N$,
SIMs with more layers are beneficial given their enhanced flexibility.

In Fig. 3, we also report the circuit complexity of the considered SIM
architectures, in terms of the number of tunable impedance components in the
circuit topology. Interestingly, we observe that 1-layer SIMs implemented with
BD-RIS are also favorable in terms of complexity, as they are slightly more
complex than 1-layer SIMs implemented with D-RIS.

Figure 3: Normalized channel gain
$G=|h|^{2}/(\|\bar{\mathbf{h}}^{R}\|^{2}\|\bar{\mathbf{h}}^{(1)}\|^{2})$ and
circuit complexity of SIM implemented with D-RIS and BD-RIS.

##  VII Conclusion

We analyze SIM-aided communication systems by using multiport network theory
and derive a physically consistent SIM-aided channel model. To understand the
role of the SIM on the derived model, we make specific assumptions to obtain a
simplified channel model that is consistent with the model employed in
previous works on SIM. With this simplified model, we compare SIM implemented
with D-RIS and BD-RIS, showing that 1-layer SIM implemented with BD-RIS
outperforms any $L$-layer SIM implemented with D-RIS.

## Appendix

Denote as $\mathbf{a}_{1}=[a_{1,1},\ldots,a_{1,N_{1}}]^{T}$ and
$\mathbf{b}_{1}=[b_{1,1},\ldots,b_{1,N_{1}}]^{T}$ the incident and reflected
waves on the first $N_{1}$ ports of the $N_{P}$-port network, respectively; as
$\mathbf{a}_{2}=[a_{2,1},\ldots,a_{2,N_{2}}]^{T}$ and
$\mathbf{b}_{2}=[b_{2,1},\ldots,b_{2,N_{2}}]^{T}$ the incident and reflected
waves on the last $N_{2}$ ports of the $N_{P}$-port network, respectively; and
as $\mathbf{a}_{3}=[a_{3,1},\ldots,a_{3,N_{3}}]^{T}$ and
$\mathbf{b}_{3}=[b_{3,1},\ldots,b_{3,N_{3}}]^{T}$ the reflected and incident
waves on the last $N_{3}$ ports of the $N_{Q}$-port network, respectively.
Given the interconnections between the two multiport networks,
$\mathbf{a}_{2}$ and $\mathbf{b}_{2}$ are also the reflected and incident
waves on the first $N_{2}$ ports of the $N_{Q}$-port network, respectively.
Thus, we have

$\begin{bmatrix}\mathbf{b}_{1}\\\
\mathbf{b}_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{P}_{11}&\mathbf{P}_{12}\\\
\mathbf{P}_{21}&\mathbf{P}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{1}\\\
\mathbf{a}_{2}\end{bmatrix},\>\begin{bmatrix}\mathbf{a}_{2}\\\
\mathbf{a}_{3}\end{bmatrix}=\begin{bmatrix}\mathbf{Q}_{11}&\mathbf{Q}_{12}\\\
\mathbf{Q}_{21}&\mathbf{Q}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{b}_{2}\\\
\mathbf{b}_{3}\end{bmatrix},$ (45)

and our goal is to characterize $\mathbf{R}$ such that

$\begin{bmatrix}\mathbf{b}_{1}\\\
\mathbf{a}_{3}\end{bmatrix}=\begin{bmatrix}\mathbf{R}_{11}&\mathbf{R}_{12}\\\
\mathbf{R}_{21}&\mathbf{R}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{a}_{1}\\\
\mathbf{b}_{3}\end{bmatrix},$ (46)

according to [10, Chapter 4]. To this end, we first use (45) to express
$\mathbf{a}_{2}$ and $\mathbf{b}_{2}$ as functions of $\mathbf{a}_{1}$ and
$\mathbf{b}_{3}$ as

$\mathbf{a}_{2}=\left(\mathbf{I}-\mathbf{Q}_{11}\mathbf{P}_{22}\right)^{-1}\mathbf{Q}_{11}\mathbf{P}_{21}\mathbf{a}_{1}+\left(\mathbf{I}-\mathbf{Q}_{11}\mathbf{P}_{22}\right)^{-1}\mathbf{Q}_{12}\mathbf{b}_{3},$
(47)
$\mathbf{b}_{2}=\left(\mathbf{I}-\mathbf{P}_{22}\mathbf{Q}_{11}\right)^{-1}\mathbf{P}_{21}\mathbf{a}_{1}+\left(\mathbf{I}-\mathbf{P}_{22}\mathbf{Q}_{11}\right)^{-1}\mathbf{P}_{22}\mathbf{Q}_{12}\mathbf{b}_{3}.$
(48)

Then, by substituting (47) into
$\mathbf{b}_{1}=\mathbf{P}_{11}\mathbf{a}_{1}+\mathbf{P}_{12}\mathbf{a}_{2}$,
which is derived from (45), we obtain

$\mathbf{b}_{1}=\mathbf{P}_{11}\mathbf{a}_{1}+\mathbf{P}_{12}\left(\mathbf{I}-\mathbf{Q}_{11}\mathbf{P}_{22}\right)^{-1}\mathbf{Q}_{11}\mathbf{P}_{21}\mathbf{a}_{1}\\\
+\mathbf{P}_{12}\left(\mathbf{I}-\mathbf{Q}_{11}\mathbf{P}_{22}\right)^{-1}\mathbf{Q}_{12}\mathbf{b}_{3},$
(49)

and by substituting (48) into
$\mathbf{a}_{3}=\mathbf{Q}_{21}\mathbf{b}_{2}+\mathbf{Q}_{22}\mathbf{b}_{3}$,
we obtain

$\mathbf{a}_{3}=\mathbf{Q}_{21}\left(\mathbf{I}-\mathbf{P}_{22}\mathbf{Q}_{11}\right)^{-1}\mathbf{P}_{21}\mathbf{a}_{1}\\\
+\mathbf{Q}_{21}\left(\mathbf{I}-\mathbf{P}_{22}\mathbf{Q}_{11}\right)^{-1}\mathbf{P}_{22}\mathbf{Q}_{12}\mathbf{b}_{3}+\mathbf{Q}_{22}\mathbf{b}_{3},$
(50)

proving the proposition.

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# A laser interferometer accelerometer for vibration sensitive cryogenic
experiments

R.Bajpai1, T.Tomaru2, K.Yamamoto3, T.Ushiba4, N.Kimura4, T.Suzuki5, T.Yamada6,
T. Honda6 1 The Graduate University for Advanced Studies, Department of
Accelerator Science, School of High Energy Accelerator Science, High Energy
Accelerator Research Organization (KEK) Tsukuba, Ibaraki, 305-0801, Japan  2
National Astronomical Observatory of Japan, 2 Chome-21-1 Osawa, Mitaka, Tokyo,
181-8588, Japan  3 University of Toyama, 3190 Gofuku, Toyama, 930-8555, Japan
4 The University of Tokyo Institute for Cosmic Ray Research Kamioka
Observatory, Higashimozumi 238, Kamioka, Hida, Gifu, 506-1205, Japan  5 The
University of Tokyo Institute for Cosmic Ray Research, Kashiwanoha 5-1-5,
Kashiwa, Chiba, 277-8582, Japan  6 High Energy Accelerator Research
Organisation, 1-1 Oho, Tsukuba, Ibaraki, 305-0801, Japan<EMAIL_ADDRESS>
###### Abstract

Monitoring motion originating from ultra low-temperature cooling systems like
cryocoolers is important for vibration sensitive cryogenic experiments like
KAGRA. Since no commercial cryogenic accelerometers are available, we
developed a compact self-calibrating accelerometer with a Michelson
interferometer readout for cryogenic use. Change in calibration factor and
drop in interferometer output originating from temperature drop were the main
concerns which were tackled. Sensitivity of $3.38\times 10^{-11}$
m/$\sqrt{\mathrm{Hz}}$ at 1 Hz was achieved at 300 K. The accelerometer was
tested inside the KAGRA cryostat; showed stable operation down to 12 K in
0.1-100 Hz band with only 1% visibility drop. Our accelerometer can be
employed in low vibration cryogenic environment for a multitude of
applications.

Keywords: vibration, cryogenics, accelerometer, low-temperature sensors,
gravitational wave detector

##  1 Introduction

Cryogenic temperatures are needed in various high precision experiments where
thermal fluctuations can be a noise source. Mechanical cryocoolers provide a
safe, low cost and reliable way to achieve cryogenic temperature. However, the
mechanical vibrations originating from the operation of cryocoolers are a
noise source in vibration sensitive experiments like KAGRA, ILC and CUROE [1,
2, 3]. Therefore, considerable efforts have been made to develop low vibration
pulse tube cryocoolers [4]. But the development of high sensitivity cryogenic
accelerometers to monitor these vibrations and design active damping systems
is still a key task as no such devices are commercially available.

Large-scale Cryogenic Gravitational wave Telescope (KAGRA), is a second-
generation gravitational-wave detector (GWD) located in Kamioka mine, Japan.
The detector is under commissioning and will join Advanced LIGO [5], and
Advanced VIRGO [6] for a joint O4 observation run in 2022. The features that
distinguish KAGRA from other GWDs are its underground location and cryogenic
operation of the four main mirrors. The underground location provides a quiet
site with low seismic and gravity gradient noise [7, 8], while the cryogenic
operation cools the mirrors down to 20 K, reducing the thermal noises.
However, vibration originating from the cryocooler operation and structural
resonances of the cryostat can contaminate the detector sensitivity making the
quite underground location redundant. Therefore, monitoring and
characterization of the vibration inside cryostat are critical for the optimum
noise performance of KAGRA.

For the aforementioned reason, we developed a sensitive cryogenic
accelerometer and evaluated its performance using the low vibration
environment of KAGRA cryostat.

##  2 Requirements

The specifications for the cryogenic accelerometer were set based on its
application for KAGRA cryostat vibration analysis. In this section, we briefly
describe the KAGRA cooling system and then specify the requirements for the
cryogenic accelerometer. Figure 1 shows the layout of KAGRA cryogenic system.
The main test masses (23 kg sapphire mirrors) in KAGRA are suspended from a
nine-stage suspension; the bottom four stages, including the mirror, are
called ’Cryogenic Payload’ [9]. It is cooled within a double-radiation shield
cryostat using four ultra-low vibration pulse-tube cryocoolers [10]. The 1st
stage of each cryocooler is connected to the outer (80 K) shield, while 2nd
stage of two cryocoolers is connected to the inner (8 K) shield. Thermal
radiation from shields effectively cools the payload down to $\sim 100$ K. In
addition, 2nd stage of the other two cryocoolers are connected to 6N (6 Nine,
$99.9999\%$ purity) aluminium cooling bars that are in thermal contact with
the payload through thin 6N Al heat-links [11] to facilitate conduction
cooling of the test mass down to 20 K. Cryocooler vibration can couple to test
mass through these heat-links.

The requirements for the cryogenic accelerometer are:

  1. 1.

Calibration: The accelerometer needs to be self-calibrating because the
calibration factor will change with temperature, and there is no reference
accelerometer available to re-calibrate it at low temperature.

  2. 2.

Temperature: The accelerometer should be able to operate inside KAGRA
cryostat, which is typically around 11-15 K when cooling down.

  3. 3.

Sensitivity: The underground location at KAGRA provides a quiet site; typical
seismic motion being $3\times 10^{-9}$ m/$\surd$Hz at 1 Hz, which is 2 order
of magnitude smaller than seismic motion at surface. We set the minimum
sensitivity required to be below the KAGRA seismic level.

Figure 1: Schematic layout of KAGRA cryogenic system. The outer black block is
the vacuum chamber; the inner boxes are 80 K and 8 K radiation shield; PTC,
CB, HL, HLVIS, are cryocoolers, heat conductor (cooling bar), heat-links,
heat-links vibration isolation system, respectively and form the cooling
system. The cryogenic payload consists of the platform (PF), marionette (MN),
intermediate-mass (IM), test mass (TM) and their recoil masses MNR, IRM, RM.
The tiny grey rectangle shows the cryogenic accelerometer mounting position
during its testing.

##  3 Development and Setup

Our accelerometer is based on a high sensitivity seismometer developed for
seismic noise measurement at Kamioka site (the relative displacement between a
pendulum and its suspension point is measured with a Michelson Interferometer)
[12, 13]. In this section, we describe the accelerometer layout and design of
individual components.

###  3.1 Accelerometer Layout

Figure 2: Cryogenic Accelerometer: (a) Schematic layout of cryogenic
accelerometer. The LASER is introduced through an optical fiber. The black
square denotes the components (proof-mass, fixed mirror, photo-diode,
beamsplitter and coil) which are housed inside the cryostat. (b) Photo of
cryogenic accelerometer in KAGRA cryostat. The laser diode and circuit box are
housed outside the cryostat. (c) Block diagram of cryogenic accelerometer in
locked state.

Based on our requirements, we designed an accelerometer with a Michelson
Interferometer (MI) readout that operates in a closed-loop. Figure 2 shows the
schematic layout of the cryogenic accelerometer. The MI is formed by two
mirrors (M1 and M2), M1 glued to a proof mass and M2 fixed rigidly to the base
plate; the laser is introduced into the cryostat through an optical fiber. The
interference signal, readout by a photo-diode, detects the relative
displacement between ground and proof mass. This signal is used as feedback to
coil magnet actuators that lock the proof mass in linear range of the
interferometer. Below unity gain frequency, the force from actuators to proof
mass is proportional to the inertial vibration of the base plate (”proof-
mass”-”beamsplitter” direction). Vibration can be derived from the feedback
signal.

Figure 2 shows the block diagram of the cryogenic accelerometer where
$H_{\mathrm{{P}}},H_{\mathrm{{I}}},H_{\mathrm{{F}}},H_{\mathrm{{D}}},H_{\mathrm{{C}}}$
are proof-mass, interferometer, filter, coil driver and coil transfer
functions, respectively; $V_{\mathrm{{1}}}$ and $V_{\mathrm{{2}}}$ are the
interferometer output and coil input (feedback signal), respectively. The
inertial force that acts on the proof-mass ($m$) for some ground vibration
($X(\omega)$) is $m\omega^{2}X(\omega)$, and the feedback force from the
actuator to lock the mass is F. From the block diagram, the expression of
vibration can be derived as:

$\displaystyle\centering|X(\omega)|=\frac{1}{m\omega^{2}}\left|\frac{1+G}{G}\right||H_{\mathrm{{C}}}||V_{\mathrm{{2}}}|\@add@centering$
(1)

where
$G=H_{\mathrm{{P}}}H_{\mathrm{{I}}}H_{\mathrm{{F}}}H_{\mathrm{{D}}}H_{\mathrm{{C}}}$
is the open-loop transfer function.

In equation (1), $|H_{\mathrm{{C}}}|\left|{1+G}/{G}\right|$ is the calibration
factor ($C$) and converts the voltage $V_{\mathrm{{2}}}$ to the feedback
force. Dividing the feedback force with proof-mass weight ($m$) gives us the
inertial acceleration of the base plate. Displacement spectral density is
obtained from the acceleration spectrum by dividing by $\omega^{2}[=(2\pi
f)^{2}]$. Finally, the expression for vibration can be simplified to:

$\displaystyle\centering|X(\omega)|=\frac{C}{m\omega^{2}}|V_{\mathrm{{2}}}|\@add@centering$
(2)

###  3.2 Cryogenic Compatibility

Cooling down accelerometer will cause attenuation at interferometer output due
to: (a) reduction in PD responsitivity and (b) signal loss from a drop in
fringe visibility originating from angular misalignment due to thermal shrink
of each component; this can cause the accelerometer to be inoperable at low
temperatures; to minimize such loss, individual components were selected and
tested.

####  3.2.1 Optics

The interference signal was read out using Thorlabs FGA21 photo-diode [14]
(active area: $\phi 2$ mm), which is known to be compatible at cryogenic
temperature through our testing. Further cooling tests showed that this
cryogenic compatibility is wavelength dependent; the results are shown in
figure 3. Note that for measurement with 1330 nm laser final temperature of PD
was 32 K; no drop in efficiency was observed at lower temperatures when the PD
was cooled with accelerometer setup in section 4.2 and 4.4 down to 16 K and 12
K, respectively.

Figure 3: Photo-diode efficiency (FGA21) as a function of Temperature, for
1310 nm (blue line) and 1550 nm (red dashed line) light.

Based on the photo-diode performance, a 15 mW, 1310 nm LASER diode was used as
the light source and introduced into the cryostat through a single-mode fiber
coupled to a collimator (beam waist-0.7 mm).

A 10 mm beamsplitter (splitting ratio: 47% (transmitted) to 46.7% (reflected))
and two ½” fused silica mirrors (reflectivity$\sim 99.5\%$) form the MI with
arm-length of 9 mm.

####  3.2.2 Mechanical Scheme

Mirror misalignment due to thermal shrink of accelerometer components can
cause a drop in fringe visibility. The mechanical components were designed to
minimize the misalignment.

  * •

Proof Mass: The proof mass is a cube of mass, $m=0.555$ kg made of Oxygen-Free
Copper (OFC), which is suspended from an OFC frame by 4, Beryllium Copper
(BeCu) blade springs of length, $l=15$ mm and thickness, $t=100\ \mu$m, which
are shown in figure 4. The resonant frequency is 9.5 Hz. A much lower resonant
frequency can be achieved using an inverted pendulum of the same size;
however, this rigid proof-mass design reduces the impact of radiation shield
tilt on fringe visibility as it restricts the motion of proof-mass in the
direction parallel to the mounting surface. Also, the proof mass makes it
possible to extend the scheme in vertical direction measurement with minor
changes.

Figure 4: Proof mass for cryogenic accelerometer. The mass is a cube of side
0.4 m made of Oxygen-Free Copper (OFC). It is suspended from an OFC frame by
4, Beryllium Copper (BeCu) blade springs of length, $l=15$ mm and thickness,
$t=100\ \mu$m.

  * •

Fixed Mirror Mount: A large contribution to visibility drop is expected to
come from the shrink of fixed mirror mount; Thorlabs, ’POLARIS-K05 mount [15]
(shown in figure 2) was selected as our cooling test showed it to have a small
angular drift of approximately $0.04^{\circ}$ and $0.05^{\circ}$, along pitch
and yaw axis respectively. In this cooling test, the mount with a mirror was
cooled down inside a cryostat; a laser reflected off the mirror falls on a
position-sensitive detector (PSD). The drift of this reflected beam is
monitored as the mount is cooled. Figure 5 shows the result of mirror mount
tilt; drift observed for pitch axis is due to non-uniform cooling of the mount
and recovers once a stable temperature of 16 K is achieved. To precisely align
the interferometer, fiber-coupled collimator was also mounted on this mount,
as shown in figure 2.

Figure 5: Result of fixed mirror mount cooling test.

The components were mounted rigidly on a base plate of dimensions: $170\times
170\times 15$ mm3 supported by 3-M6 screws to establish three-point contact.
With proper alignment, fringe visibility of $98.5\%$ was achieved. From the
results of PD efficiency and fixed mirror mount drift measurement, we expect
the attenuation at interferometer output will not be an issue at cryogenic
temperature; this was confirmed with a cooling test as reported in section
4.2.

####  3.2.3 Electronics

Electronics for accelerometer are housed outside the cryostat, as seen in
figure 2. A Lake Shore temperature sensor (DT-670) is bolted to the base plate
to monitor the accelerometer temperature.

The current signal from the PD is converted to voltage signal and passed
through a differential amplifier to remove the dc offset, generating the error
signal. The error signal is fed to the controller circuit (band-pass filter +
coil driver), tuning the unity gain frequency of the open-loop transfer
function (G) above 100 Hz and generating the actuating current signal. This
signal drives coil magnet actuators, locking the proof-mass. The Samarium
Cobalt (SmCo) magnets are fixed to the back of the proof-mass with Epoxy (3MTM
Scotch WeldTM DP-190). SmCo magnets are used as their magnetism doesn’t change
significantly between the room and cryogenic temperature [16].

###  3.3 Calibration

The calibration factor (C) will change as the accelerometer is cooled because:
(a) actuator efficiency will be lower due to reduction in magnetism of SmCo
magnets and change in the coupling factor between the magnet and coil, (b)
optical loss at interferometer output from a drop in fringe visibility and (c)
reduction in PD responsitivity. However, the accelerometer can be calibrated
at any temperature by measuring open-loop transfer function ($G$) and coil
transfer function ($H_{C}$) as follows:

  * •

Open-loop transfer function($G$) is measured at interferometer output by swept
sine excitation.

  * •

Coil transfer function ($H_{\mathrm{{C}}}$) shown in figure 2 is calculated
from coil to interferometer transfer function
$(V_{\mathrm{{1}}}/V_{\mathrm{{2}}})$ as:

$\displaystyle
H_{\mathrm{{C}}}=\frac{(V_{\mathrm{{1}}}/V_{\mathrm{{2}}})}{H_{\mathrm{{I}}}H_{\mathrm{{P}}}}$
(3)

where, $H_{\mathrm{{P}}}$ is the proof-mass transfer function and is
approximated to $1/m(2\pi f_{\mathrm{{r}}})^{2}$ for frequency, $f\ll
f_{\mathrm{{r}}}$ (resonant frequency).

$H_{\mathrm{{I}}}$ is the interferometer transfer function.

  * •

Interferometer transfer function can be calculated when there is no fed back
to the proof mass (free-swinging Michelson) as:

$\displaystyle H_{\mathrm{I}}=\frac{4\pi V_{\mathrm{1free}}}{\lambda}$ (4)

where,

$\lambda=1310$ nm is the LASER wavelength

$V_{\mathrm{1free}}$ is the amplitude of interferometer output when the proof
mass is free

We call this ’wavelength calibration’ and can be used to calibrate the
accelerometer at any temperature. Figure 6 shows the calibration factor
measured at room temperature and atmospheric pressure.

Figure 6: Calibration factor of cryogenic accelerometer determined using
wavelength calibration. The interferometer transfer function,
$H_{\mathrm{I}}$, was $16.54$ nm/V, and the coil transfer function
$H_{\mathrm{C}}$ was $0.0019$ N/V.

##  4 Testing

The accelerometer was tested at room and cryogenic temperature to confirm it
satisfies the requirements set in section 2. In this section we report the
results of these tests and accelerometer performance.

###  4.1 Performance Comparison

To confirm the reliability of wavelength calibration as described in section
3.3, the cryogenic accelerometer and a commercial accelerometer (RION LA-50)
were mounted (in air) together on an optical table; vibration was measured
simultaneously. The spectrum measured by the two accelerometers is consistent,
and the coherence between the output of the two is almost 1 over the entire
spectrum, as seen in figure 6. We conclude that the absolute value of our
accelerometer output is reliable.

Figure 7: Vibration spectrum of the optical table measured; by our
accelerometer (blue) and a commercial accelerometer (red) (top). Coherence
(green) between output of two accelerometers (bottom).

###  4.2 Cryogenic Testing

Next, the accelerometer was cooled down in a cryostat in our laboratory at the
High Energy Accelerator Research Organisation (KEK) to evaluate the low-
temperature performance. Initial visibility of accelerometer at 297 K was
$98.5\%$; figure 8 shows interferometer visibility as accelerometer was
cooled. The intermediate drop is linked to the pitch drift of the fixed mirror
mount and non-uniform cooling of individual components. Once the accelerometer
reached a stable temperature of 16 K, visibility recovered to $94\%$.

Finally, the accelerometer was re-calibrated, and vibration of the cryostat
was measured. We confirmed that thermal shrink did not cause any significant
visibility drop, and accelerometer shows stable operation down to 16 K.

Figure 8: Results of cryogenic testing of accelerometer. (a) Visibility of
Michelson Interferometer Vs temperature of accelerometer. (b) Vibration
spectrum of a cryostat in KEK at 16 K. The peak at 1.37 Hz and its harmonics
are from cryocooler operation.

###  4.3 Sensitivity

The main noise sources for the accelerometer are the intensity noise of the
laser and electronic noise of the circuits, photo-diode and DC supply. To
evaluate the sensitivity of accelerometer we block the proof-mass arm with a
beam dump; under such conditions, there is no interference, and the output
shows the approximate noise level ($V_{\mathrm{{noise}}}$) of the
accelerometer itself due to laser intensity and electronic noise of the DC
supply. The minimum vibration that can be measured by our accelerometer is
shown in blue in figure 9. The PD dark and circuit noise are evaluated by
measuring $V_{\mathrm{{noise}}}$ when the laser and dc supply is turned off.
Figure 9 shows that the noise level of the cryogenic accelerometer is lower
than Kamaioka seismic motion; therefore, it can be tested in KAGRA cryostat in
0.1-100 Hz bandwidth.

Figure 9: Sensitivity of cryogenic accelerometer dominated with laser
intensity and DC supply electronic (blue) compared to PD dark+ circuit noise
(green) and Kamiaoka seismic motion (red).

###  4.4 Onsite Testing

As the vibration level in KEK is 10-100 times larger than Kamioka, we tested
the cryogenic accelerometer in KAGRA cryostat, figure 10. The final
temperature of the accelerometer and inner shield was 12 K and 14.7 K,
respectively; the final interferometer visibility was $96$%. Figure 11 shows
the vibration spectrum of inner shield at $12$ K when all cryocoolers are
operating. We confirmed that our developed accelerometer can operate in low
vibration cryogenic environment.

Figure 10: Cryogenic Accelerometer (red box) mounted inside one of the KAGRA
cryostat. Figure 11: Vibration Spectra of KAGRA cryostat shown in figure 10.
At the time of measurement inner shield and accelerometer were at 14.7 K and
12 K, respectively, and all cryocoolers were operating. The seismic motion was
measured by a commercial accelerometer (RION LA-50) placed next to the
cryostat in air.

##  5 Conclusion

We developed a cryogenic accelerometer with a Michelson interferometer
readout. The mechanical and optical components were selected, designed and
tested to minimize loss of interferometer output at cryogenic temperature.
Accelerometer was tested at room and cryogenic temperature; it satisfied the
sensitivity, temperature and calibration requirements. We installed the
accelerometer in KAGRA cryostat, and it showed stable operation down to 12 K.
We conclude the high sensitivity, compactness and stable operation at
temperatures down to 12 K make our device ideal for various low vibration
cryogenic experiments.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Acknowledgment

This work was supported by MEXT, JSPS Leading-edge Research Infrastructure
Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, JSPS
Grant-in-Aid for Scientific Research on Innovative Areas 2905: JP17H06358,
JP17H06361 and JP17H06364, JSPS Core-to-Core Program A. Advanced Research
Networks, JSPS Grant-in-Aid for Scientific Research (S) 17H06133, the joint
research program of the Institute for Cosmic Ray Research, University of
Tokyo, National Research Foundation (NRF) and Computing Infrastructure Project
of KISTI-GSDC in Korea, Academia Sinica (AS), AS Grid Center (ASGC) and the
Ministry of Science and Technology (MoST) in Taiwan under grants including AS-
CDA-105-M06, the LIGO project, and the Virgo project. We would also like to
thank KEK Machine Shop for machining the mechanical components of the
accelerometer.

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