††thanks: corresponding author††thanks: corresponding author

# Light-shinning-through-thin-wall radio frequency cavities for probing dark
photon

Dmitry Salnikov<EMAIL_ADDRESS>Moscow State University, 119991
Moscow, Russia  Institute for Nuclear Research, 117312 Moscow, Russia  D. V.
Kirpichnikov<EMAIL_ADDRESS>Institute for Nuclear Research, 117312 Moscow,
Russia  Petr Satunin<EMAIL_ADDRESS>Moscow State University, 119991
Moscow, Russia  Institute for Nuclear Research, 117312 Moscow, Russia  Leysan
Valeeva<EMAIL_ADDRESS>Moscow State University, 119991 Moscow,
Russia  Institute for Nuclear Research, 117312 Moscow, Russia

###### Abstract

We address the radio frequency (RF) cavity experiment for probing dark photons
which is a modification of the light-shining-through-thin-wall (LSthinW) setup
with a relatively thin conducting barrier between cylindrical emitter and
hollow receiver. The regarding experimental facility allows to probe dark
photons effectively even in the off-shell regime, i. e. when dark photons mass
exceeds the driven frequency of the emitter cavity, that is pumped by
electromagnetic mode. We compare the sensitivity of two concrete setup
configurations: two cylindrical cavities placed to each other by an end-caps,
and nested geometry of the cavities in which the cylindrical receiver is
encapsulated into the emitter. We show that for a certain range of dark photon
mass the nested configuration can provide an enhanced sensitivity if one
compares that with a separated emitter setup.

††preprint: INR-TH-2024-002

##  I Introduction

A dark (hidden) photon is a massive hypothetical particle that interacts with
the Standard Model (SM) through kinetic mixing with a visible photon Holdom
(1986). For relatively small values of the kinetic mixing parameter, dark
photon (DP) can avoid cosmological constraints and thus it can be a viable
candidate for the dark matter Nelson and Scholtz (2011); Arias  _et al._
(2012). It is worth noticing, a hidden photon is addressed also in the
scenarios as a mediator between dark matter and visible sector Antel  _et al._
(2023); Zhevlakov  _et al._ (2023); Andreev  _et al._ (2023). Typically a DP
is described in the framework of $U(1)$ gauge extension of SM, that is natural
requirement for a string theory Okun (1982).

A variety of precise searches for hidden photon have been carried out recently
in numerous laboratory and accelerator based experiments. For instance, a
searches for hidden photons with relatively heavy masses (in the
$\lesssim\mathcal{O}(1)~{}\mbox{GeV}$ range) have utilized $e^{+}e^{-}$ Lees
_et al._ (2017) and $pp$ colliders Aaij  _et al._ (2018), the NA64 fixed
target Banerjee  _et al._ (2020, 2018), the JUNO underground detectors D’Eramo
_et al._ (2023), and FASER beam-dump Abreu  _et al._ (2024) experiments.

In addition, we also address the low mass limits (sub
$\sim\mathcal{O}(1)~{}\mbox{eV}$ range) derived from tests of Coulomb’s law
Williams  _et al._ (1971); Abel  _et al._ (2008) and atomic spectroscopy
Jaeckel and Roy (2010). The limits on dark photons that can be hypothetical DM
candidates are performed in haloscope searches Gelmini  _et al._ (2020); Liu
_et al._ (2022); Chiles  _et al._ (2022); Chen  _et al._ (2024); Knirck  _et
al._ (2023); Chaudhuri  _et al._ (2015); Alesini  _et al._ (2023); McAllister
_et al._ (2024); Tang  _et al._ (2023), gravitational wave detector Ismail
_et al._ (2023), cryogenic millimeter-wave cavity Kotaka  _et al._ (2023),
dish antenna Horns  _et al._ (2013), radio telescopes An  _et al._ (2021),
gaseous detectors Graesser  _et al._ (2024), photonic chip Blinov  _et al._
(2024), astrophysical environments Brahma  _et al._ (2023), and Earth-based
magnetometers Bloch and Kalia (2020); Fedderke  _et al._ (2021) (for recent
review see also Ref. Berlin  _et al._ (2022a) and references therein).

For completeness, it is worth noticing that a dark photons for the
sufficiently small masses below $\lesssim 1~{}\mbox{eV}$ have been probed with
the SM photon regeneration facilities such as DarkSRF Romanenko  _et al._
(2023), CROWS Betz  _et al._ (2013), ADMX Wagner  _et al._ (2010), and
microwave cavity experiment Povey  _et al._ (2010). The latter facilities are
referred to light-shining-through-wall (LSW) experiments. To be more specific,
the regarding detection schemes to search for dark photons are associated with
electromagnetically isolated conducting cavities tuned to the same frequency
Van Bibber  _et al._ (1987); Jaeckel and Ringwald (2008). In particular, a
dark photons produced from high-density SM photons in the emitter cavity
propagate in space until reaching the other receiver hollow cavity where they
convert back into SM photons.

The authors of Ref. Graham  _et al._ (2014) argued that, while previous
literature Betz  _et al._ (2013); Povey  _et al._ (2010); Jaeckel and Ringwald
(2008); Kim  _et al._ (2021); Parker  _et al._ (2013a, b) focused on the
production and detection of transverse modes of hidden photon, the
longitudinal mode allows a significant improvement in LSW experimental
sensitivity. In the present paper we employ the formalism of Ref. Graham  _et
al._ (2014) in order to calculate the sensitivity of cylindrical receiver that
is nested (or encapsulated) inside the emitter cavity Kim  _et al._ (2021)
(see e. g. Fig. 1 for detail) for sufficiently thin wall between receiver and
emitter Berlin  _et al._ (2023). The similar experimental setup is addressed
in Ref. Berlin  _et al._ (2023) as light-shining-through-thin-wall (LSthinW)
but for distant cylindrical cavities (see e. g. Fig. 2 for detail). The
typical barrier between endcaps of the receiver and emitter cavity Berlin  _et
al._ (2023) is sufficiently small $d\simeq\mathcal{O}(10)~{}\mu\mbox{m}\ll
1/\omega$, where
$\omega\sim\mathcal{O}(1)~{}\mbox{GHz}\sim~{}\mathcal{O}(1)~{}\mu\mbox{eV}$ is
a typical driven radio frequency (RF) of the emitter pump mode (it matches
with a signal EM mode frequency that is resonantly induced in the hollow
receiver).

The LSthinW setup allows to enhance the sensitivity in the off-shell DP
region, i. e. when hidden photon mass, $m_{A^{\prime}}$, is relatively large
$m_{A^{\prime}}\gg\omega$. In addition, employing the sufficiently low driven
frequency of the emitter, one can achieve a large density of the source
photons and high quality of the superconducting RF (SRF) cavity Berlin  _et
al._ (2023).

In the present paper, we estimate the impact on the DP sensitivity that is
associated with the wall thickness $d$ between cavities for the nested design
of LSthinW setup shown in Fig. 1. We also show that there is an advantage of
nested receiver as opposed to its distant location Fig. 2 for the specific
resonator design.

This paper is organized as follows. In Sec. II we discuss general properties
of DP scenario and derive regarding equations of motion. In Sec. III we derive
a signal power that implies probing the DP in the SRF receiver. In Sec. IV we
calculate overlapping factor for distant location of receiver and emitter,
while in Sec. V we derive the form-factor expression for the cylindrical
receiver that is nested in the emitter. In Sec. VI we estimate the sensitivity
of the aforementioned LSthinW designs. In Appendix A and B we specify some
useful formulas for the form-factor derivation in the analytical form.

##  II Framework of the scenario

Let us consider the Lagrangian, describing a coupling of the ordinary Standard
Model (SM) photon, $\hat{A}_{\mu}$, with a massive vector hidden state,
$\hat{A}_{\mu}^{\prime}$, through the kinetic mixing

$\mathcal{L}\supset-\frac{1}{4}\hat{F}_{\mu\nu}^{2}-\frac{1}{4}\left(\hat{F}^{\prime}_{\mu\nu}\right)^{2}+\frac{1}{2}m_{A^{\prime}}^{2}\left(\hat{A}^{\prime}_{\mu}\right)^{2}+\frac{\epsilon}{2}\hat{F}_{\mu\nu}^{\prime}\hat{F}^{\mu\nu}-J_{\mu}\hat{A}^{\mu},$
(1)

where we exploit the hat over the boson fields in order to label the states of
the benchmark interaction Eq. (1) in so called field basis Graham  _et al._
(2014) or kinetically mixed basis Fedderke  _et al._ (2021), $\epsilon$ is a
dimensionless coupling of the kinetic mixing, that is assumed to be small,
$\epsilon\ll 1$, $\hat{F}_{\mu\nu}^{(\prime)}$ is the stress tensor of the
(dark) photon, $J^{\mu}$ is the typical $U(1)$ electromagnetic current
density. Now let us get rid of the non-diagonal kinetic mixing term
$\propto\epsilon$ in Eq. (1) by the following replacement of the field basis
states $\hat{A}_{\mu}^{(\prime)}$ to the mass basis states
$A_{\mu}^{(\prime)}$,

$\hat{A}_{\mu}=A_{\mu}+\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}A_{\mu}^{\prime},\quad\hat{A}^{\prime}_{\mu}=\frac{1}{\sqrt{1-\epsilon^{2}}}A^{\prime}_{\mu},$
(2)

as a result, one gets the following redefined Lagrangian

$\displaystyle\mathcal{L}\supset-\frac{1}{4}F_{\mu\nu}^{2}-\frac{1}{4}\left(F^{\prime}_{\mu\nu}\right)^{2}+\frac{1}{2}\frac{m_{A^{\prime}}^{2}}{(1-\epsilon^{2})}\left(A^{\prime}_{\mu}\right)^{2}$
$\displaystyle-
J_{\mu}\left(A^{\mu}+\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}A^{\prime\mu}\right),$
(3)

that leads to the decoupled equations of motions for both $A_{\mu}$ and
$A_{\mu}^{\prime}$

$\partial^{2}_{\mu}A_{\nu}=J_{\nu},$ (4)
$\partial^{2}_{\mu}A^{\prime}_{\nu}+\frac{m_{A^{\prime}}^{2}}{1-\epsilon^{2}}A_{\nu}^{\prime}=\frac{\epsilon}{(1-\epsilon^{2})}J_{\nu},$
(5)

where we impose the Lorentz gauge $\partial_{\mu}A^{\mu}=0$ and on-shell Proca
condition $\partial_{\mu}A^{\prime\mu}=0$, implying that it leads to
conservation of the electromagnetic current $\partial_{\mu}J^{\mu}=0$ (see e.
g. Ref. Fedderke  _et al._ (2021) for detail). The electric and magnetic
fields are defined in a usual manner for both SM, $A_{\mu}$, and hidden
photon, $A_{\mu}^{\prime}$,

${\bf E}^{(\prime)}=-\bm{\nabla}\phi^{(\prime)}-\partial_{t}{\bf
A}^{(\prime)},\qquad{\bf B}^{(\prime)}=\bm{\nabla}\times{\bf A}^{(\prime)},$
(6)

where we imply the typical four-vector notation via potential,
$A^{(\prime)0}\equiv\phi^{(\prime)}$ and vector terms $A^{(\prime)i}\equiv{\bf
A}^{(\prime)}$. However, it was pointed out in Ref. Graham  _et al._ (2014)
that in order to calculate the physical state propagation one should express
the fields in the mass basis through the linear combination of the visible,
$A^{\mu}_{\text{vis}}$, and invisible, $A^{\mu}_{\text{inv}}$, modes in the
following form Berlin  _et al._ (2023)

$\displaystyle A^{\mu}=\sqrt{1-\epsilon^{2}}A^{\mu}_{\text{vis}}-\epsilon
A^{\mu}_{\text{inv}},$ (7) $\displaystyle A^{\prime\mu}=\epsilon
A^{\mu}_{\text{vis}}+\sqrt{1-\epsilon^{2}}A^{\mu}_{\text{inv}}.$ (8)

As a result, for the regarding basis, the equations of motions (4) and (5) can
be rewritten as follows

$\displaystyle\left(\partial^{2}_{t}-\bm{\nabla}^{2}\right)A^{\mu}_{\text{vis}}=J^{\mu}_{\text{SM}}-\frac{\epsilon
m_{A^{\prime}}^{2}}{(1-\epsilon^{2})}A^{\prime\mu}$ (9)
$\displaystyle\left(\partial^{2}_{t}-\bm{\nabla}^{2}+\frac{m_{A^{\prime}}^{2}}{(1-\epsilon^{2})}\right)A^{\mu}_{\text{inv}}=-\frac{\epsilon
m_{A^{\prime}}^{2}}{(1-\epsilon^{2})}A^{\mu},$ (10)

where we use convention for the effective SM density current
$J^{\mu}_{\text{SM}}\equiv\sqrt{1-\epsilon^{2}}J^{\mu}$. In Eqs. (9) and (10)
we keep the dependence on $A^{\prime\mu}$ and $A^{\mu}$ in the right hand side
respectively in order to emphasize, that at the leading order approach,
$\epsilon\ll 1$, the visible component $A^{\mu}_{\text{vis}}$ is sourced by
the electromagnetic SM current $J^{\mu}_{\text{SM}}$. Moreover, the invisible
mode $A^{\mu}_{\text{inv}}$ is sourced only by the typical effective current
$\propto-\epsilon m_{A^{\prime}}^{2}A^{\mu}$, that is subdominant to the
$J^{\mu}_{\text{SM}}$.

Figure 1: Typical sketch of the cavity setup with the encapsulated receiver.
Figure 2: Typical scheme of the setup for distant cavities. We note that a
realistic configuration implies thin conducting barrier between cavities such
that $d\ll R,L$.

##  III Cavity response

Let us consider benchmark experimental setups consisting of the high-quality
superconducting RF cylindrical receiver cavity which is

  * •

separated from the emitter by the thin barrier of $d\ll R$, where
$R=R_{{\text{em}}}=R_{{\text{rec}}}$ is a typical radius of the cavities (see
Fig. 2);

  * •

nested into emitter such that they have common perfectly conducting thin side
wall of thickness $d\ll R$ (see Fig. 1).

Let x and y be the coordinates associated with the emitter and receiver frame
respectively. In addition, let l be the vector linking the origins of the
frames of both emitter and receiver for the coaxial design of the experimental
setup. In this case one has a simple link between them
${\text{\bf{x}}}={\text{\bf{l}}}+{\text{\bf{y}}}$.

The first cavity is pumped by a single electromagnetic mode (EM)

$\displaystyle{\bf E}({\text{\bf{x}}},t)={\bf
E}_{{\text{em}}}({\text{\bf{x}}})e^{-i\omega t},$ (11) $\displaystyle{\bf
B}({\text{\bf{x}}},t)={\bf B}_{{\text{em}}}({\text{\bf{x}}})e^{-i\omega t},$
(12)

oscillating at frequency $\omega$ and serves as an emitter of dark photon. In
(11) and (12) we employ convention of Hill (2014) for the time dependence
$\propto e^{-i\omega t}$ in order to match the notation of cavity eigenmodes
Hill (2014). The second cavity is a quiet hollow receiver of dark photon, thus
it is expected that the detecting EM mode is resonantly exited due to
regeneration of hidden vector state radiated from the emitter cavity.

Now let us define the invisible electric field ${\bf
E}_{{\text{inv}}}\equiv{\bf E}^{\prime}-\epsilon{\bf E}$, this means that Eq.
(10) implies the following massive wave equation

$\left(\partial^{2}_{t}-\bm{\nabla}^{2}+m_{A^{\prime}}^{2}\right){\bf
E}_{\text{inv}}=-\epsilon m_{A^{\prime}}^{2}{\bf E}.$ (13)

The Eq. (13) means that the invisible electric field inside the receiver
cavity sourced by the massless field of the emitter ${\bf E}={\bf
E}_{\text{em}}e^{-i\omega t}$. At the order of $\mathcal{O}(\epsilon)$ one has
${\bf E}_{\text{inv}}\simeq{\bf E}^{\prime}$ and thus the dark photon electric
mode in the receiver can be expressed via Green’s function in the following
form Graham  _et al._ (2014)

${\bf E}^{\prime}({\text{\bf{x}}},t)=-\frac{\epsilon
m_{A^{\prime}}^{2}}{4\pi}\int\limits_{V_{{\text{em}}}}\\!\\!d^{3}{\text{\bf{x}}}^{\prime}\,\frac{e^{-i\omega
t+ik_{A^{\prime}}|{\text{\bf{x}}}-{\text{\bf{x}}}^{\prime}|}}{|{\text{\bf{x}}}-{\text{\bf{x}}}^{\prime}|}{\bf
E}_{\text{em}}({\text{\bf{x}}}^{\prime}),$ (14)

where integration $d^{3}{\text{\bf{r}}}^{\prime}$ is performed over the
emitter cavity volume $V_{\text{em}}$ for the emitter frame, $k_{A^{\prime}}$
is a wavenumber that reads as
$k_{A^{\prime}}\equiv\sqrt{\omega^{2}-m_{A^{\prime}}^{2}}$ for relatively
small masses $m_{A^{\prime}}\lesssim\omega$ and
$k_{A^{\prime}}=i\varkappa_{A^{\prime}}$ with $\varkappa_{A^{\prime}}$ being
$\varkappa_{A^{\prime}}\equiv\sqrt{m_{A^{\prime}}^{2}-\omega^{2}}$ for heavy
hidden vector state $m_{A^{\prime}}\gtrsim\omega$.

Now let us estimate the resonant response of the receiver cavity. We define
first the visible electric field as ${\bf E}_{\text{vis}}\equiv{\bf
E}+\epsilon{\bf E}^{\prime}$ that obeys the following boundary condition on
the conducting surface

$\left[{\text{\bf{n}}}\times{\bf E}_{\text{vis}}\right]\big{|}_{\partial
V_{{\text{rec}}}}=0,$ (15)

where the unit vector n is defined along the normal to the interior bound
$\partial V_{{\text{rec}}}$ of the receiver cavity Graham  _et al._ (2014).
One can show Graham  _et al._ (2014) that at leading order
$\mathcal{O}(\epsilon)$ the receiver response ${\bf E}_{\text{vis}}$ sourced
by the time derivative of the typical effective current

$\bm{\nabla}\times(\bm{\nabla}\times{\bf E}_{\text{vis}})+\partial^{2}_{t}{\bf
E}_{\text{vis}}=-\partial_{t}{\bf J}_{{\text{eff}}}({\text{\bf{y}}},t),$ (16)

where the left hand side equations imply the derivatives with respect to the
receiver frame y, $\partial_{t}{\bf
J}_{{\text{eff}}}({\text{\bf{x}}},t)=-i\omega{\bf
J}_{{\text{eff}}}({\text{\bf{x}}})e^{-i\omega t}$, here the stationary part of
the regarding current reads as follows

${\bf
J}_{{\text{eff}}}({\text{\bf{x}}})=\frac{\epsilon}{i\omega}\left[\bm{\nabla}\cdot\left(\bm{\nabla}\cdot{\bf
E}^{\prime}({\text{\bf{x}}})\right)-m_{A^{\prime}}^{2}{\bf
E}^{\prime}({\text{\bf{x}}})\right],$ (17)

where ${\bf E}^{\prime}({\text{\bf{x}}})$ is a spatial profile term of Eq.
(14). The solution to Eq. (16) for the receiver can be expressed in the
following form

$\displaystyle{\bf E}_{\text{vis}}({\text{\bf{y}}},t)$ $\displaystyle\simeq$
$\displaystyle-\frac{Q}{\omega_{n}}{\bf
E}^{(n)}_{{\text{rec}}}({\text{\bf{y}}})e^{-i\omega t}$ (18)
$\displaystyle\times$
$\displaystyle\\!\\!\frac{\int_{V_{\text{rec}}}\\!d^{3}\\!{\text{\bf{y}}}^{\prime}\left({\bf
E}^{(n)*}_{\text{rec}}({\text{\bf{y}}}^{\prime})\cdot{\bf
J}_{{\text{eff}}}({\text{\bf{y}}}^{\prime})\right)}{\int_{V_{{\text{rec}}}}d^{3}{\text{\bf{y}}}^{\prime}|{\bf
E}_{\text{rec}}^{(n)}({\text{\bf{y}}}^{\prime})|^{2}},$

where ${\bf E}^{(n)}_{\text{rec}}({\text{\bf{y}}})$ is divergence-free vacuum
cavity-mode, $\bm{\nabla}\cdot{\bf E}^{(n)}_{\text{rec}}({\text{\bf{y}}})=0$,
$Q$ is a quality factor of the receiver cavity, $\omega_{n}$ is a cavity
eigen-frequency that is closed to the driven frequency
$\omega_{n}\simeq\omega$ in the resonant response regime, i. e. when the
signal build up time $t$ is sufficiently large $t\gg Q/\omega$. The quality
factor is chosen to be $Q\simeq 4\times 10^{10}$ throughout the paper. We
notice again for completeness, that the integration over
$d^{3}{\text{\bf{y}}}^{\prime}$ in Eq. (18) is performed over the receiver
cavity volume $V_{{\text{rec}}}$ in the receiver frame
${\text{\bf{y}}}^{\prime}$.

One can show Berlin  _et al._ (2022b) that the total energy stored in the
receiver cavity due to the resonant response is

$W_{{\text{rec}}}\simeq\frac{1}{2}\int\limits_{V_{\text{rec}}}d^{3}{\text{\bf{y}}}\left[\langle\mbox{Re}[{\bf
B}_{\text{vis}}({\text{\bf{y}}},t)]^{2}\rangle_{t}+\langle\mbox{Re}[{\bf
E}_{\text{vis}}({\text{\bf{y}}},t)]^{2}\rangle_{t}\right]$ (19)

where we employ the time averaging notations $\langle...\rangle_{t}$ for the
energy density of both the magnetic, $\propto\langle\mbox{Re}[{\bf
B}_{\text{vis}}({\text{\bf{y}}},t)]^{2}\rangle_{t}/2$, and electric,
$\propto\langle\mbox{Re}[{\bf
E}_{\text{vis}}({\text{\bf{y}}},t)]^{2}\rangle_{t}/2$ fields in the perfect
conductor, here $\mbox{Re}[...]$ is a real part of the EM fields. Remarkably,
the regarding energy densities are equal in the resonant regime Berlin  _et
al._ (2022b), and the resulted expression for $W_{\text{rec}}$ can be
expressed only through the integral over the electric energy density
$\propto|{\bf E}_{\text{vis}}({\text{\bf{y}}})|^{2}/2$ term, where ${\bf
E}_{\text{vis}}({\text{\bf{y}}})$ is a stationary part of Eq. (18).

From the discussion above one can estimate the signal power emission of a
cavity

$P_{{\text{sign}}}\simeq\frac{\omega}{Q}W_{rec}=\frac{\omega}{2Q}\int\limits_{V_{\text{rec}}}d^{3}{\text{\bf{y}}}|{\bf
E}_{{\text{vis}}}({\text{\bf{y}}})|^{2}=\frac{Q}{2\omega}\big{|}\mathcal{G}\big{|}^{2},$
(20)

where $\mathcal{G}$ is a overlapping factor (here we keep it dimensional for
brevity)

$\mathcal{G}=\frac{\int_{V_{\text{rec}}}d^{3}{\text{\bf{y}}}\left({\bf
E}^{(n)*}_{\text{rec}}({\text{\bf{y}}})\cdot{\bf
J}_{{\text{eff}}}({\text{\bf{y}}})\right)}{\left(\int_{V_{{\text{rec}}}}d^{3}{\text{\bf{y}}}|{\bf
E}_{\text{rec}}^{(n)}({\text{\bf{y}}})|^{2})\right)^{1/2}}.$ (21)

It is worth noticing Eq. (21) implies that the signal is agnostic to overall
constant normalization pre-factor of the receiver mode, ${\bf
E}_{\text{rec}}^{(n)}({\text{\bf{y}}})$, however it depends on the specific
mode type and the overlapping integrals in the numerator and denominator of
Eq. (21).

We estimate the sensitivity as an maximum output in the receiver that is
defined by the Dickie radiometer equation Dicke (1946),

$\mbox{SNR}=\frac{P_{\text{sign}}}{P_{\text{noise}}}\sqrt{t\Delta\nu},$ (22)

where $t$ is an integration time of the signal, $\Delta\nu$ is a signal
bandwidth, $P_{\text{noise}}$ is a power of thermal noise that is estimated to
be $P_{\text{noise}}\simeq T\Delta\nu$ for $\omega\ll T$. The typical
temperature of the receiver is chosen to be $T\simeq 4~{}\mbox{K}$. It is
worth noticing that the narrowest possible bandwidth can be as small as
$\Delta\nu\simeq 1/t$, thus in the numerical estimation we conservatively set
integration time to be $t\simeq 1~{}\mbox{year}$.

Let us present the general formula to calculate the sensitivity for the mixing
parameter $\epsilon$ for the specific cavity geometry,

$\epsilon=\left(\frac{2T\cdot\mbox{SNR}}{Q\omega^{3}(E^{0}_{\rm
em})^{2}V^{2}_{\rm em}V_{\rm
rec}}\right)^{\frac{1}{4}}\left(\frac{\Delta\nu}{t}\right)^{\frac{1}{8}}m_{A^{\prime}}^{-1}|\widetilde{{\cal
G}}|^{-1/2}.$ (23)

where $\widetilde{{\cal G}}$ is a dimensionless geometric factor,

$\displaystyle\widetilde{{\cal G}}=\dfrac{1}{\omega^{3}}$
$\displaystyle\int\limits_{V_{\rm rec}}\dfrac{d^{3}{\text{\bf{y}}}}{V_{\rm
rec}}\int\limits_{V_{\rm em}}\dfrac{d^{3}{\text{\bf{x}}}}{V_{\rm em}}\,{\cal
E}_{\rm rec}^{*i}({\text{\bf{y}}})\,{\cal E}_{\rm
em}^{j}({\text{\bf{x}}})\times$ (24)
$\displaystyle\times(m^{2}_{A^{\prime}}\delta_{ij}-\partial_{{\text{\bf{y}}}i}\partial_{{\text{\bf{y}}}j})\left\\{-\dfrac{\exp(ik_{A^{\prime}}|{\text{\bf{x}}}-{\text{\bf{y}}}-{\text{\bf{l}}}|)}{4\pi|{\text{\bf{x}}}-{\text{\bf{y}}}-{\text{\bf{l}}}|}\right\\},$

where ${\cal E}^{i}_{{\text{rec}}}({\text{\bf{y}}})$ and ${\cal
E}^{i}_{{\text{rec}}}({\text{\bf{x}}})$ are cavity modes of the receiver and
emitter respectively which are connected with electric fields as follows

$\displaystyle E_{{\text{em}}}^{i}({\text{\bf{x}}})=E^{0}_{\rm em}\cdot{\cal
E}_{\rm em}^{i}({\text{\bf{x}}}),$ (25) $\displaystyle
E_{{\text{rec}}}^{i}({\text{\bf{y}}})={\cal
E}_{{\text{rec}}}^{i}({\text{\bf{y}}}),$ (26)

where $E_{0}^{\text{em}}$ is a magnitude of the driven emitter electric field,
indices $i$ and $j$ label the $i^{\text{th}}$ and $j^{\text{th}}$ components
of the electric field respectively. In the present paper we set the typical
driving EM field to be
$E_{0}^{\text{em}}=30~{}\mbox{MV}/\mbox{m}=0.1~{}\mbox{T}$ (see e. g. Ref.
Berlin  _et al._ (2023) for detail). In addition we introduce the pre-factor
$1/\omega^{3}$ in Eq. (24) in order to obtain dimensionless form-factor
$\widetilde{\cal G}$. In order to estimate the expected reach of the
experimental configuration, we put $\mbox{SNR}\simeq 5$ in Eq. (23).

In addition we note that eigenmodes are normalized as follows

$\int\limits_{V_{{\text{rec}}/{\text{em}}}}\\!\\!\\!\\!d^{3}{\text{\bf{r}}}\
|{\cal
E}^{i}_{{\text{rec}}/{\text{em}}}({\text{\bf{r}}})|^{2}=V_{{\text{rec}}/{\text{em}}}.$
(27)

In Eq. (26) we omit the amplitude of the of receiver field for brevity (see e.
g. discussion after Eq. (21) for details).

##  IV Coaxial distant cavities: TM010 mode

In this section we consider the experimental design of two coaxial distant
cavities shown in Fig. 2. The vector linking the cavity frames would be then
${\text{\bf{l}}}=(0,0,l_{z})$. For that notation the distance between closest
endcaps of the cavities is $d=l_{z}-L$. We consider the TM010 mode of the
receiver and emitter. This mode implies the following non-zero electric field
components along $z$-direction

$\displaystyle{\cal
E}_{z,{\text{em}}}^{\text{TM}_{010}}(\rho,\phi,z)=J_{0}\left(k_{1\rho}\rho\right)/J_{1}(\omega
R),$ (28) $\displaystyle{\cal
E}_{z,{\text{rec}}}^{\text{TM}_{010}}(\rho^{\prime},\phi^{\prime},z^{\prime})=J_{0}\left(k_{1\rho}\rho^{\prime}\right)/J_{1}(\omega
R),$ (29)

where $(\rho,\phi,z)$ and $(\rho^{\prime},\phi^{\prime},z^{\prime})$ are the
cylindrical coordinates associated with emitter and receiver frame
respectively, $k_{1\rho}$ is ratio $k_{1\rho}=x_{01}/R$ with $x_{01}$ being a
first zero of Bessel function, $J_{0}(x_{01})=0$. We take into account in Eqs.
(28) and (29) that tangential component of the electric field is zero on the
conducting surface (15).

It is worth noticing that employing k momentum representation of the spherical
Green’s function

$\dfrac{e^{ik_{A^{\prime}}|{\bf x-y-{\text{\bf{l}}}}|}}{4\pi|{\bf
x-y-{\text{\bf{l}}}}|}=-\int\dfrac{d^{3}{\bf k}}{(2\pi)^{3}}\,\dfrac{e^{-i{\bf
k}({\bf x-y-{\text{\bf{l}}}})}}{{\bf
k}^{2}-(k^{2}_{A^{\prime}}+i\varepsilon)},$ (30)

one can rewrite overlapping factor involving the nonzero effective current
component $J^{z}_{\text{eff}}({\text{\bf{x}}})\neq 0$ in Eq. (21) in the
following form

$\displaystyle\widetilde{\mathcal{G}}=\frac{1}{\omega^{3}V_{\text{rec}}V_{{\text{em}}}}\\!\\!\int\limits_{V_{\text{rec}}}\\!\\!\\!d^{3}{\text{\bf{y}}}\\!\\!\\!\int\limits_{V_{\text{em}}}\\!\\!d^{3}{\text{\bf{x}}}\frac{d^{3}{\text{\bf{k}}}}{(2\pi)^{3}}\times$
$\displaystyle\times{\cal
E}_{z,{\text{rec}}}^{*\text{TM}_{010}}({\text{\bf{y}}}){\cal
E}_{z,{\text{em}}}^{\text{TM}_{010}}({\text{\bf{x}}})\frac{(k_{z}^{2}+m_{A^{\prime}}^{2})e^{-i{\text{\bf{k}}}({\text{\bf{x}}}-{\text{\bf{y}}}-{\text{\bf{l}}})}}{\left[{\text{\bf{k}}}^{2}-k_{A^{\prime}}^{2}-i\varepsilon\right]},$
(31)

where we imply that the integration over $d^{3}{\text{\bf{k}}}$ is performed
for cylindrical frame, in addition we note that a small complex contribution
from $+i\varepsilon$ in the denominator of Eq. (31) is crucial for spherical
wave Fourier transformation in Eq. (14). Remarkably, the 9D integral for the
overlapping factor (31) can be reduced to the simplified form

$\widetilde{\mathcal{G}}=\frac{L^{2}R^{2}}{\omega
V_{\text{rec}}V_{{\text{em}}}}\times\mathcal{I},$ (32)

where $\mathcal{I}$ can be expressed through the 2D integral

$\displaystyle\mathcal{I}$ $\displaystyle=$
$\displaystyle\\!\\!\int\limits_{0}^{\infty}\\!\\!dk_{\rho}k_{\rho}\frac{J_{0}^{2}(k_{\rho}R)}{\left(k_{\rho}^{2}-\omega^{2}\right)^{2}}$
(33) $\displaystyle\times$
$\displaystyle\\!\\!\\!\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!\\!dk_{z}\frac{4\sin^{2}\left(k_{z}L/2\right)(k_{z}^{2}+m_{A^{\prime}}^{2})e^{ik_{z}l_{z}}}{L^{2}k_{z}^{2}\left(k_{z}^{2}-\left[k_{A^{\prime}}^{2}-k_{\rho}^{2}+i\varepsilon\right]\right)}.$

In order to obtain Eq. (33) from Eq. (31) one can exploit the helpful
integrals that are collected in Appendix A. Furthermore, the calculation of
the second term in Eqs. (33) is straightforward for $d\equiv l_{z}-L\gtrsim
0$, since the $k_{z}$-complex integration contour can be closed upward in
order to pick up pole in this case. In particular, for
$k_{A^{\prime}}^{2}\equiv\omega^{2}-m_{A^{\prime}}^{2}\gtrsim 0$, the
integration over $k_{z}$ splits into a sum of two terms,
$\mathcal{I}=\mathcal{I}[k_{\rho}\gtrsim
k_{A^{\prime}}]+\mathcal{I}[k_{\rho}\lesssim k_{A^{\prime}}]$, where

$\displaystyle\mathcal{I}[k_{\rho}\gtrsim k_{A^{\prime}}]$ $\displaystyle=$
$\displaystyle 2\pi
k_{A^{\prime}}\int\limits_{0}^{\infty}dx\frac{J_{0}^{2}\left(k_{A^{\prime}}R\sqrt{1+x^{2}}\right)}{\left(\omega^{2}-k_{A^{\prime}}^{2}(1+x^{2})\right)}$
$\displaystyle\times$
$\displaystyle\frac{4\sinh^{2}\left(xk_{A^{\prime}}L/2\right)}{2\left(k_{A^{\prime}}xL\right)^{2}}\exp\left(-xk_{A^{\prime}}l_{z}\right)$

is a term that is associated with the typical region $k_{\rho}\gtrsim
k_{A^{\prime}}$ implying the variable redefinition
$k_{\rho}=k_{A^{\prime}}(1+x^{2})^{1/2}$ in the integrand, and

$\displaystyle\mathcal{I}[k_{\rho}\lesssim k_{A^{\prime}}]\\!\\!$
$\displaystyle=2\pi ik_{A^{\prime}}\\!\\!\\!\\!\\!$
$\displaystyle\int\limits_{0}^{1}\\!\\!dx\frac{J_{0}^{2}\left(k_{A^{\prime}}R\sqrt{1-x^{2}}\right)}{\left(\omega^{2}-k_{A^{\prime}}^{2}(1-x^{2})\right)}$
$\displaystyle\times\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\frac{4\sin^{2}\left(xk_{A^{\prime}}L/2\right)}{2\left(k_{A^{\prime}}xL\right)^{2}}\exp\left(ixk_{A^{\prime}}l_{z}\right)$

is a term that corresponds to the $k_{z}$ integration with the typical
parametric area $k_{\rho}\lesssim k_{A^{\prime}}$ for the variable replacement
$k_{\rho}=k_{A^{\prime}}(1-x^{2})^{1/2}$. Moreover, for TM010 mode both Eqs.
(IV) and (IV) imply that $\omega=k_{1\rho}\equiv x_{01}/R$. For
$\varkappa_{A^{\prime}}^{2}\equiv m_{A^{\prime}}^{2}-\omega^{2}\gtrsim 0$ the
overlapping factor simply reads as

$\displaystyle\mathcal{I}$ $\displaystyle=2\pi\varkappa_{A^{\prime}}$
$\displaystyle\int\limits_{1}^{\infty}dx\frac{J_{0}^{2}\left(\varkappa_{A^{\prime}}R\sqrt{x^{2}-1}\right)}{\left(\omega^{2}-\varkappa_{A^{\prime}}^{2}(x^{2}-1)\right)}$
(36) $\displaystyle\times$
$\displaystyle\frac{4\sinh^{2}\left(x\varkappa_{A^{\prime}}L/2\right)}{2\left(\varkappa_{A^{\prime}}xL\right)^{2}}\exp\left(-x\varkappa_{A^{\prime}}l_{z}\right),$

where we use the integration variable replacement
$k_{\rho}=\varkappa_{A^{\prime}}(x^{2}-1)^{1/2}$. The asymptotic of
$\widetilde{\cal G}$ in the sufficiently large masses region
$m_{A^{\prime}}\gg 1/d\gtrsim\omega$ for the distant cavity design is
estimated to be (see e. g. Appendix B for detail)

$\widetilde{\cal
G}\propto\dfrac{e^{-m_{A^{\prime}}d}}{(m_{A^{\prime}})^{7/2}},$ (37)

as a result of Eq. (37) one has

$\epsilon\propto(m_{A^{\prime}})^{3/4}\times e^{\frac{m_{A^{\prime}}d}{2}}$
(38)

for the distant setup.

##  V Coaxial encapsulated cavity: TM010 mode

In this section we consider the following design of the light-shining-through-
wall experimental setup shown in Fig. 1. The emitter is space between internal
and external coaxial cylindrical hollow superconducting cavities of radius
$R_{2}$ and $R_{3}$ respectively. The receiver is a internal hollow
cylindrical cavity of radius $R_{1}$ that is encapsulated into emitter. Both
external and internal cavities have an equal length $L$, moreover the origins
of their frames coincide, such that, one should set ${\text{\bf{l}}}\equiv 0$
in Eq. (21). We suppose that the thickness of the superconducting wall between
cavities Berlin  _et al._ (2023) is sufficiently small $d\equiv R_{2}-R_{1}\ll
R_{3}\simeq\mathcal{O}(10)\mbox{cm}$. We note that Eqs. (15) and (29) imply
that the electric field mode of the cylindrical receiver is given by

${\cal
E}_{z,{\text{rec}}}^{\text{TM}_{010}}(\rho^{\prime},\phi^{\prime},z^{\prime})=J_{0}\left(\omega\rho^{\prime}\right)/J_{1}(\omega
R_{1}),$ (39)

where $\omega=x_{01}/R_{1}$ is a eigen-frequency of TM010 mode. The emitter
electric field can be expressed through the specific cylindrical function
$Z_{0}(x)$ in the following form,

$\displaystyle{\cal E}_{z,{\text{em}}}^{\text{TM}_{010}}$
$\displaystyle\\!\\!(\rho,\phi,z)$ $\displaystyle=$
$\displaystyle\beta\left(J_{0}(\omega\rho)-\dfrac{J_{0}(\omega
R_{2})}{Y_{0}(\omega R_{2})}\cdot Y_{0}(\omega\rho)\right)\equiv\beta
Z_{0}(\omega\rho),$

where $Y_{0}(x)$ is the Neumann function of zeroth order. The pre-factor
$\beta$ in Eq. (V) can be obtained from the normalization integral Eq. (27),

$\beta=\left[\dfrac{R^{2}_{3}-R^{2}_{2}}{R^{2}_{3}Z_{1}^{2}(\omega
R_{3})-R^{2}_{2}Z_{1}^{2}(\omega R_{2})}\right]^{1/2},$ (41)

where $R_{3}$ is a solution to the equation $Z_{0}(\omega R_{3})\equiv 0$, in
Eq. (41) we also exploit additional cylindrical function in the following form
$Z_{1}(x)\equiv J_{1}(x)-Y_{1}(x)J_{0}(\omega R_{2})/Y_{0}(\omega R_{2})$.

Remarkably, for above mentioned encapsulated cavity design the overlapping
factor reads

$\widetilde{\mathcal{G}}=\widetilde{\mathcal{G}}_{3}-\widetilde{\mathcal{G}}_{2},$
(42)

where $\widetilde{{\cal G}}_{i}$ are defined for $i=2,3$ as follows

$\widetilde{\mathcal{G}}_{i}=\frac{L^{2}\beta^{2}R_{1}R_{i}Z_{1}(\omega
R_{i})}{\omega V_{\text{rec}}V_{{\text{em}}}}\times\mathcal{I}_{i},$ (43)

here the typical integral $\mathcal{I}_{i}$ can be expressed in the following
form

$\displaystyle\mathcal{I}_{i}$ $\displaystyle=$
$\displaystyle\\!\\!\int\limits_{0}^{\infty}\\!\\!dk_{\rho}k_{\rho}\frac{J_{0}(k_{\rho}R_{1})J_{0}(k_{\rho}R_{i})}{\left(k_{\rho}^{2}-\omega^{2}\right)^{2}}$
(44) $\displaystyle\times$
$\displaystyle\\!\\!\\!\\!\\!\int\limits_{-\infty}^{\infty}\\!\\!\\!dk_{z}\frac{4\sin^{2}\left(k_{z}L/2\right)(k_{z}^{2}+m_{A^{\prime}}^{2})}{L^{2}k_{z}^{2}\left(k_{\rho}^{2}-\left[k_{A^{\prime}}^{2}-k_{z}^{2}+i\varepsilon\right]\right)}.$

Next, by employing Eqs. (52) and (53) in Eq. (44) for the case of
$k_{A^{\prime}}^{2}\equiv\omega^{2}-m_{A^{\prime}}^{2}\gtrsim 0$, the
integration over $k_{\rho}$ can be split into a sum of two terms,
$\mathcal{I}_{i}=\mathcal{I}_{i}[k_{z}\gtrsim
k_{A^{\prime}}]+\mathcal{I}_{i}[k_{z}\lesssim k_{A^{\prime}}]$, where

$\displaystyle\mathcal{I}_{i}[k_{z}\\!\\!$ $\displaystyle\\!\\!\lesssim\\!\\!$
$\displaystyle\\!\\!k_{A^{\prime}}]=i\pi
k_{A^{\prime}}\int\limits_{0}^{1}dx\,\left[\dfrac{\sin(k_{A^{\prime}}xL/2)}{(k_{A^{\prime}}xL/2)}\right]^{2}$
$\displaystyle\times$
$\displaystyle\dfrac{J_{0}(k_{A^{\prime}}R_{1}\sqrt{1-x^{2}})H^{(1)}_{0}(k_{A^{\prime}}R_{i}\sqrt{1-x^{2}})}{k_{A^{\prime}}^{2}x^{2}+m^{2}_{A^{\prime}}}$

is a term for the typical integration region $k_{z}\lesssim k_{A^{\prime}}$in
Eq. (44), and

$\displaystyle\mathcal{I}_{i}[k_{z}\\!\\!$ $\displaystyle\\!\\!\gtrsim\\!\\!$
$\displaystyle\\!\\!k_{A^{\prime}}]=2k_{A^{\prime}}\int\limits_{1}^{\infty}dx\,\left[\dfrac{\sin(k_{A^{\prime}}xL/2)}{(k_{A^{\prime}}xL/2)}\right]^{2}$
$\displaystyle\times$
$\displaystyle\dfrac{I_{0}(k_{A^{\prime}}R_{1}\sqrt{x^{2}-1})K_{0}(k_{A^{\prime}}R_{i}\sqrt{x^{2}-1})}{k_{A^{\prime}}^{2}x^{2}+m^{2}_{A^{\prime}}}$

is a term that is associated with the integration range $k_{z}\gtrsim
k_{A^{\prime}}$ in Eq. (44). We note that both Eqs. (V) and (V) imply the
variable redefinition $k_{z}=xk_{A^{\prime}}$.

Finally, for $\varkappa_{A^{\prime}}^{2}\equiv
m_{A^{\prime}}^{2}-\omega^{2}\gtrsim 0$ one has

$\displaystyle\\!\\!\mathcal{I}_{i}\\!\\!$
$\displaystyle=2\varkappa_{A^{\prime}}\int\limits_{0}^{\infty}dx\,\left[\dfrac{\sin(\varkappa_{A^{\prime}}xL/2)}{(\varkappa_{A^{\prime}}xL/2)}\right]^{2}$
(47) $\displaystyle\times$
$\displaystyle\dfrac{I_{0}(\varkappa_{A^{\prime}}R_{1}\sqrt{x^{2}+1})K_{0}(\varkappa_{A^{\prime}}R_{i}\sqrt{x^{2}+1})}{\varkappa_{A^{\prime}}^{2}x^{2}+m^{2}_{A^{\prime}}},$

where we redefine the typical variable $k_{z}=\varkappa_{A^{\prime}}x$. In
addition, an analytical form of Eq. (42) can be calculated for
$m_{A^{\prime}}\gg\omega$ and $R_{1}\gg d$. In particular, the asymptotic
estimation of $\widetilde{{\cal G}}$ in the approach
$\varkappa_{A^{\prime}}L\gg 1$ implies that $(\pi
ax^{2})^{-1}\sin^{2}ax\simeq\delta(x)$ as long as $a\gg 1$, therefore this
results in

$\widetilde{\cal G}\propto\dfrac{e^{-m_{A^{\prime}}d}}{m^{3}_{A^{\prime}}}.$
(48)

Next, by using an estimate (48) and (23), we obtain the typical dependence of
the expected sensitivity in the following form

$\epsilon\propto(m_{A^{\prime}})^{1/2}\times e^{\frac{m_{A^{\prime}}d}{2}}$
(49)

for the encapsulated setup. Remarkably, Eq. (49) implies that for encapsulated
setup the dependence of $\epsilon$ on mass $m_{A^{\prime}}$ is proportional to
$\propto(m_{A^{\prime}})^{1/2}$ as opposed to $\propto(m_{A^{\prime}})^{3/4}$
for the separated receiver setup, see Eq. (38).

In order to validate our results, we calculated numerically the six-
dimensional integral for the overlapping factor defined by Eq. (24) in the
typical regions and the parameter spaces for both distant and encapsulated
receiver. The latter numerical estimates coincide fairly well with the semi-
analytical results (43) and (42).

##  VI Results and discussion

Figure 3: The setup sensitivity to dark photon parameters for TM010 mode. Top
panel: distant configuration, bottom panel: encapsulated configuration. The
solid and dashed lines represent the typical parameters $R=R_{3}=5\,{\rm
cm},L=20\,{\rm cm}$ and $R=R_{3}=10\,{\rm cm},L=40\,{\rm cm}$, respectively.
Green lines correspond to the typical barrier thickness of
$d=1000~{}\mu\mbox{m}$, blue lines are associated with $d=400~{}\mu\mbox{m}$,
and red lines show the ultimate limits for $d=10~{}\mu\mbox{m}$. We set the
typical setup parameters to be $Q=4\times 10^{10}$,
$E^{0}_{\text{em}}=30\mbox{MV}/\mbox{m}$, $T=4~{}\mbox{K}$, and $t\simeq
1~{}\mbox{year}$. We also show current experimental regions that have been
already ruled out by DarkSRF Pathfinder Romanenko  _et al._ (2023), CROWS Betz
_et al._ (2013), and XENON1T An  _et al._ (2020), these bounds were adapted
from Ref. Berlin  _et al._ (2023)

In Fig. 3 we show the sensitivities to the dark photon coupling $\epsilon$ and
mass $m_{A^{\prime}}$, for two different choices of the experimental
configurations: distant receiver (top panel) and encapsulated receiver (bottom
panel). The expected reaches are shown for the barrier thickness in the range
$10~{}\mu\mbox{m}\lesssim d\lesssim 1~{}\mbox{mm}$. The code for numerical
calculations and drawing figures can be shared upon request.

Given the benchmark geometric configuration ($R\lesssim 10~{}\mbox{cm}$,
$L\lesssim 40~{}\mbox{cm}$) for distant setup the resonant enhancement of the
DP sensitivity $\epsilon\lesssim 4\times 10^{-13}$ can be achieved for the
typical masses $5\times 10^{-6}~{}\mbox{eV}\lesssim m_{A^{\prime}}\lesssim
10^{-5}~{}\mbox{eV}$. The expected reach for off-shell regime,
$m_{A^{\prime}}\gg\omega$, in this case can be as small as $\epsilon\lesssim
5\times 10^{-10}$ for the masses below $m_{A^{\prime}}\lesssim
10^{-3}~{}\mbox{eV}$.

The nested design of the emitter for the radius $R_{3}\lesssim 10~{}\mbox{cm}$
and the typical length $L\lesssim 40~{}\mbox{cm}$ can provide the peak
sensitivity at the level of $\epsilon\lesssim 2\times 10^{-13}$ for resonant
mass region $10^{-5}~{}\mbox{eV}\lesssim m_{A^{\prime}}\lesssim 2\times
10^{-5}~{}\mbox{eV}$. The ultimate thickness of the barrier between emitter
and receiver, $d\simeq 10~{}\mu\mbox{m}$, can ensure the off-shell DP,
$m_{A^{\prime}}\gg\omega$, sensitivity at the level of $\epsilon\lesssim
10^{-11}$ for the mass range below $m_{A^{\prime}}\lesssim
10^{-2}~{}\mbox{eV}$.

It is shown in Fig. 3 that the LSthinW sensitivity bound in case of off-shell
dark photons, $m_{A^{\prime}}\gtrsim\omega$ can be associated with two typical
regimes. The first one is associated with the mass range
$m_{A^{\prime}}\gtrsim d^{-1}$ that relates to the exponential decreasing of
the sensitivity. The second one is the intermediate regime for $\omega\lesssim
m_{A^{\prime}}\lesssim d^{-1}$. In this mass range the sensitivity is still
high, and the typical $\epsilon$ increases as a power low function on the dark
photon mass. The slope of the curves in the encapsulated configuration is
flatter than in the distant one. Thus, we can conclude that the encapsulated
receiver setup can provide an enhanced sensitivity for the case of large dark
photon mass and the narrow wall between cavities.

The possible extension of the present work can be addressed to a probing of
axion-like particles Salnikov  _et al._ (2023); Gao and Harnik (2021); Berlin
_et al._ (2022a) in LSthinW setup for both distant and encapsulated receiver.
We leave that task for future work.

##  VII Acknowledgements

The work on analytic and semi-analytic calculation of the setup sensitivity is
supported by RSF grant no. 21-72-10151.

##  Appendix A Auxiliary integrals

In this section we summarize some helpful integrals for calculation of the
form-factor $\widetilde{\cal G}$. The first one is an indefinite integral for
the product of the Bessel functions Gradshteyn and Ryzhik (1943)

$\displaystyle\int$ $\displaystyle\\!\\!x\,dx\,z_{p}(\alpha x)\zeta_{p}(\beta
x)$ $\displaystyle=$ $\displaystyle\dfrac{z_{p}(\alpha x)\beta
x\zeta_{p-1}(\beta x)-\zeta_{p}(\beta x)\alpha xz_{p-1}(\alpha
x)}{\alpha^{2}-\beta^{2}},$

where $\alpha$ and $\beta$ are arbitrary real numbers, $z_{p}(x)$ and
$\zeta_{p}(x)$ are arbitrary Bessel functions of $p^{\rm th}$ order. The
second one is the integral over the angular coordinate

$\int\limits^{2\pi}_{0}d\phi\exp[-i(k_{x}\rho\cos\phi+k_{y}\rho\sin\phi)]=2\pi
J_{0}(k_{\rho}\rho),$ (51)

where $k_{\rho}=(k_{x}^{2}+k_{y}^{2})^{1/2}$.

One can show that for sufficiently thin wall between emitter and receiver of
the encapsulated design, $d\ll R_{1}$, the following estimate for the integral
holds

$\displaystyle\int\limits^{\infty}_{0}$
$\displaystyle\frac{k_{\rho}dk_{\rho}}{(k_{\rho}^{2}-(q^{2}+i\varepsilon))}\cdot\frac{J_{0}(k_{\rho}R_{1})J_{0}(k_{\rho}R_{i})}{(k_{\rho}^{2}-\omega^{2})^{2}}$
(52) $\displaystyle=$
$\displaystyle\frac{i\pi}{2}\frac{J_{0}(qR_{1})H^{(1)}_{0}(qR_{i})}{(q^{2}-\omega^{2})^{2}},$

where $H^{(1)}_{0}(qR_{i})(x)$ is a Hankel function of first kind,
$q^{2}=k_{A^{\prime}}^{2}-k_{z}^{2}$, the indices are $i=2,3$, the regarding
radii $R_{i}$ are be estimated to be $R_{2}\simeq R_{1}$ and $R_{3}\simeq
R_{1}x_{02}/x_{01}$, with $x_{01}\simeq 2.405$ and $x_{02}\simeq 5.520$ being
a first and second zero of Bessel $J_{0}(x)$, such that
$J_{0}(x_{01})=J_{0}(x_{02})=0$. For pure imaginary $q\equiv
iQ=i(\varkappa_{A^{\prime}}^{2}+k_{z}^{2})^{1/2}$ the integration yields

$\displaystyle\int\limits^{\infty}_{0}$
$\displaystyle\dfrac{k_{\rho}dk_{\rho}}{k_{\rho}^{2}+Q^{2}-i\varepsilon}\cdot\dfrac{J_{0}(R_{1}k_{\rho})J_{0}(R_{i}k_{\rho})}{(k_{\rho}^{2}-\omega^{2})^{2}}$
(53) $\displaystyle=$
$\displaystyle\dfrac{I_{0}(R_{1}Q)K_{0}(R_{i}Q)}{(Q^{2}+\omega^{2})^{2}},$

where we imply that $H^{(1)}_{0}(ix)=\dfrac{2}{i\pi}K_{0}(x)$ and
$J_{0}(ix)=I_{0}(x)$ with $K_{0}(x)$ and $I_{0}(x)$ being a Macdonald and
Infeld cylindrical (modified Bessel) functions respectively Gradshteyn and
Ryzhik (1943).

##  Appendix B Asymptotic for distant cavity design

Implying the limits $m_{A^{\prime}}\gg\omega$, $m_{A^{\prime}}R\gg 1$, and
$d\ll L$ for Eq. (53) one can show that Eq. (33) reads

$\mathcal{I}\simeq\int\limits_{-\infty}^{+\infty}\frac{dk_{z}}{2R}\dfrac{1}{(k_{z}^{2}+m_{A^{\prime}}^{2})^{3/2}}\left[\dfrac{\sin\frac{k_{z}L}{2}}{\frac{k_{z}L}{2}}\right]^{2}e^{ik_{z}(d+L)},$
(54)

where we imply that $\varkappa_{A^{\prime}}\simeq m_{A^{\prime}}$, the above
mentioned integral can be rewritten in the following form

$\displaystyle\\!\\!\mathcal{I}\\!$ $\displaystyle\\!\simeq\\!$
$\displaystyle\\!\\!\dfrac{2}{RL^{2}m_{A^{\prime}}^{4}}$
$\displaystyle\\!\\!\times\\!\\!$
$\displaystyle\\!\\!\left[I(m_{A^{\prime}}(d\\!+\\!2L))\\!+\\!I(m_{A^{\prime}}d)\\!-\\!2I(m_{A^{\prime}}(d\\!+\\!L))\right]\\!,$

where $I(a)$ is a typical combination of the Macdonald, $K_{n}(a)$, and
Struve, $L_{n}(a)$, functions Gradshteyn and Ryzhik (1943)

$\displaystyle I(a)\\!$ $\displaystyle=$
$\displaystyle-1\\!+\\!\dfrac{a^{2}}{2}\\!\Bigl{[}K_{2}(a)\\!-\\!K_{0}(a)\\!$
(56) $\displaystyle+$
$\displaystyle\\!\dfrac{\pi}{2}K_{0}(a)L_{-1}(a)\\!+\\!\dfrac{\pi}{2}K_{1}(a)L_{0}(a)\Bigr{]}.$

By expanding Eq. (B) for large argument $m_{A^{\prime}}d\gg 1$ and employing
Eq. (56) one can obtain Gradshteyn and Ryzhik (1943)

$\mathcal{I}\\!\simeq\dfrac{e^{-m_{A^{\prime}}d}}{RL^{2}m_{A^{\prime}}^{4}}\left(\frac{\pi
m_{A^{\prime}}d}{2}\right)^{1/2}\left(1+\mathcal{O}\left(\frac{1}{m_{A^{\prime}}d}\right)\right),$
(57)

that leads to Eq. (37) for the asymptotic of the overlapping factor in case of
separated cavities.

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