††thanks: Equal contribution††thanks: Equal contribution††thanks:
Corresponding author

# Duality-symmetric axion electrodynamics and haloscopes of various geometries

Dai-Nam Le<EMAIL_ADDRESS>Atomic Molecular and Optical Physics Research
Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho
Chi Minh City 700000, Vietnam  Faculty of Applied Sciences, Ton Duc Thang
University, Ho Chi Minh City 700000, Vietnam  Le Phuong Hoang
<EMAIL_ADDRESS>Square Lab, Hanoi University of Science and
Technology, Hanoi 100000, Viet Nam  Binh Xuan Cao<EMAIL_ADDRESS>Square Lab, Hanoi University of Science and Technology, Hanoi 100000, Viet Nam
School of Mechanical Engineering, Hanoi University of Science and Technology,
Ha Noi 100000, VietNam

###### Abstract

Within the dual symmetric point of view, the theory for seeking axion dark
matter via haloscope experiments is derived by exactly solving the dual
symmetric axion electrodynamics equation. Notwithstanding that the
conventional theory of axion electrodynamics presented in [Phys. Rev. Lett.
51, 1415 (1983) and J. Phys. A: Math. Gen. 19, L33 (1986)] is more commonly
used in haloscope theory, we show that the dual symmetric axion
electrodynamics has more advantages to apply into haloscope theory. First, the
dual symmetric and conventional perspective of axion electrodynamics coincide
under long-wavelength approximation. Moreover, dual symmetric theory can
obtain an exact analytical expression of the axion-induced electromagnetic
field for any states of axion. This solution has been used in conventional
theory for long-wavelength approximation. The difference between two theories
can occur in directional axion detection or electric sensing haloscopes. For
illustrative purposes, we consider the various type of resonant cavities:
cylindrical solenoid, spherical solenoid, two-parallel-sheet cavity, toroidal
solenoid with a rectangular cross-section, and with a circular cross-section.
The resonance of the axion-induced signal as well as the ratio of the energy
difference over the stored energy inside the cavity are investigated in these
types of cavity.

Axion electrodynamics, dual symmetry, haloscope, resonant cavity.

##  I Introduction

It has been long time stated that dark matter (DM) plays an important role on
the gravitational evolution of our Universe, especially from $10^{4}$ to
$10^{10}$ years after Big Bang [1]. While there are myriad convincing
observations of DM in astrophysical level, the evidences that supports the
existence of DM in particle level are still insufficient. On the effort to
fulfill the Standard Model (SM) of particle physics, several theories and
experimental schemes have been proposed to detect this exotic matter. Axion is
one of well-known candidates employed to interpret the existence of cold DM
(CDM) [2, 3, 4, 5] which implies the DM velocity much smaller than light
speed. This cold axion DM is a hypothetical pseudoscalar particle resulted
from Peccei-Quinn (PQ) symmetry linked to the absence of charge-parity (CP)
violationswith strong interactions [6, 7, 8]. Modern experimental schemes of
axion detection utilize the interaction between axion and two photons which is
described as axion haloscope searches [9, 10]. The theory that explains the
coupling between axion field and electromagnetic field is so-called axion
electrodynamics and has been studied for a long time [11, 12, 13]. The current
axion haloscope experiments employ conventional axion electrodynamics proposed
by P. Sikivie in which the minimal SM is modified to include axion coupled
term [9]. However, the set of modified Maxwell equations derived from Sikivie
assumptions does not fulfill the duality preservation which is the intrinsic
symmetry of Maxwell equations from the invariance under $SO(2)$ [14, 15].
Alternatively, L. Visinelli proposed a new set of Maxwell equations
considering both axions and magnetic monopoles which preserves the duality
relation. Both sets of equations proposed by P. Sikivie and L. Visinelli are
derived from same axion electromagnetic Lagrangian [12] as the following:

$\mathcal{L}_{tot}=\mathcal{L}_{Maxwell}+\mathcal{L}_{a}+\mathcal{L}_{U},$ (1)

where
$\mathcal{L}_{Maxwell}=-(4\mu_{0})^{-1}F^{\mu\nu}F_{\mu\nu}-A_{\mu}J_{e}^{\mu}-B_{\mu}J_{m}^{\mu}$
is the classical Maxwell electromagnetic Lagrangian,
$\mathcal{L}_{a}=(4\mu_{0})^{-1}g_{a\gamma\gamma}aF^{\mu\nu}\tilde{F}_{\mu\nu}=-(4\mu_{0}c)^{-1}g_{a\gamma\gamma}a\mathbf{E}\cdot\mathbf{B}$
is axion-biphoton coupling term,
$\mathcal{L}_{U}=2^{-1}\left[(\partial^{\mu}a)(\partial_{\mu}a)-\omega_{a}^{2}a^{2}\right]$
is axion self Lagrangian which preserves the duality relation under the
substitution of axion coupled term, $\omega_{a}$ is the angular frequency of
the axion ’s oscillation and $g_{a\gamma\gamma}$ is axion-biphoton coupling
constant [16]. Here $J_{e}^{\mu}$ and $J_{m}^{\mu}$ are four-current density
associated with electric and magnetic charges and $A_{\mu}$, $B_{\mu}$ are
their corresponding four-potentials [17, 12, 18]. From equation (1), the new
set of modified Maxwell equations proposed by L. Visinelli is manifested as
the following [12, 13, 19, 20]:

$\left(\square+\dfrac{m_{a}^{2}c^{2}}{\hbar^{2}}\right)a=\dfrac{2g_{a\gamma\gamma}}{\mu_{0}c}\mathbf{E}\cdot\mathbf{B},$
(2)
$\displaystyle\pmb{\nabla}\cdot\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)=$
$\displaystyle\dfrac{\rho_{e}}{\varepsilon_{0}},$ (3a)
$\displaystyle\pmb{\nabla}\cdot\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)=$
$\displaystyle c\mu_{0}\rho_{m},$ (3b)
$\displaystyle\pmb{\nabla}\times\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)=$
$\displaystyle-\dfrac{\partial\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)}{c\partial
t}-\mu_{0}\mathbf{J}_{m},$ (3c)
$\displaystyle\pmb{\nabla}\times\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)=$
$\displaystyle\dfrac{\partial\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)}{c\partial
t}+c\mu_{0}\mathbf{J}_{e},$ (3d)

whereas $\square=c^{-2}\partial_{t}\partial_{t}-\pmb{\nabla}^{2}$ is
d’Alembert operator and the mass of axion $m_{a}$ links to its frequency
$\omega_{a}$ by $m_{a}c^{2}=\hbar\omega$.

Noticeably, in References [12, 13], these equations (3) had been shown to be
equivalent to the conventional axion electrodynamics equations in References
[9, 11]

$\displaystyle\pmb{\nabla}\cdot\mathbf{E}-cg_{a\gamma\gamma}\mathbf{B}\cdot\pmb{\nabla}a=$
$\displaystyle\dfrac{\rho_{e}}{\varepsilon_{0}}$ (4a)
$\displaystyle\pmb{\nabla}\cdot\mathbf{B}=$ $\displaystyle 0$ (4b)
$\displaystyle\pmb{\nabla}\times\mathbf{E}=$
$\displaystyle-\dfrac{\partial\mathbf{B}}{\partial t}$ (4c)
$\displaystyle\pmb{\nabla}\times\mathbf{B}-\frac{g_{a\gamma\gamma}}{c}\mathbf{E}\times\pmb{\nabla}a=$
$\displaystyle\dfrac{\partial\mathbf{E}}{c^{2}\partial
t}-\frac{g_{a\gamma\gamma}\mathbf{B}}{c}\dfrac{\partial a}{\partial
t}+\mu_{0}\mathbf{J}_{e},$ (4d)

when the axion field and the magnetic monopole sources meet the following
conditions

$\displaystyle\rho_{m}$
$\displaystyle=cg_{a\gamma\gamma}a\rho_{e}+\dfrac{g_{a\gamma\gamma}}{\mu_{0}c}\left(\mathbf{E}+cg_{a\gamma\gamma}a\mathbf{B}\right)\cdot\pmb{\nabla}a$
(5a) $\displaystyle\mathbf{J}_{m}$
$\displaystyle=cg_{a\gamma\gamma}a\mathbf{J}_{e}-\dfrac{g_{a\gamma\gamma}}{c\mu_{0}}\dfrac{\partial
a}{\partial t}\left(\mathbf{E}+cg_{a\gamma\gamma}a\mathbf{B}\right)$ (5b)
$\displaystyle\quad\quad\quad\quad\quad-\dfrac{g_{a\gamma\gamma}}{\mu_{0}}\left(c\mathbf{B}-g_{a\gamma\gamma}a\mathbf{E}\right)\times\pmb{\nabla}a.$

Therefore, in the viewpoint of duality conservation, the conventional axion
electrodynamics equations (4) are valid if and only if there is magnetic
monopole sources $\rho_{m}\neq 0,\mathbf{J}_{m}\neq 0$. Inversely, when there
is no magnetic monopole sources $\rho_{m}=0,\mathbf{J}_{m}=0$, the axion
electrodynamics equations becomes [12, 13, 19, 20]

$\displaystyle\pmb{\nabla}\cdot\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)$
$\displaystyle=$ $\displaystyle\dfrac{\rho_{e}}{\varepsilon_{0}}$ (6a)
$\displaystyle\pmb{\nabla}\cdot\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)$
$\displaystyle=$ $\displaystyle 0$ (6b)
$\displaystyle\pmb{\nabla}\times\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)$
$\displaystyle=$
$\displaystyle-\dfrac{\partial\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)}{c\partial
t}$ (6c)
$\displaystyle\pmb{\nabla}\times\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)$
$\displaystyle=$
$\displaystyle\dfrac{\partial\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)}{c\partial
t}+c\mu_{0}\mathbf{J}_{e}.$ (6d)

It is noticed that the set of conventional axion electrodynamics equation (4)
is derived from the Lagrangian principle with the Lagrangian given in (1) when
neglecting the magnetic four-current term i.e $B_{\mu}J^{\mu}_{m}$.

In this paper, we achieve the explicit solutions of these equations (6) in
various geometries of haloscope cavities. We demonstrate that the duality
symmetric theory is valid as it comes up with the same solutions with
conventional axion electrodynamics proposed by P. Sikivie in the long-
wavelength regime. Furthermore, this analytical solution can be used to
describe the electromagnetic fields inside the haloscope cavities without
long-wavelength approximation (LWA) in the duality symmetric theory. It means
that the theory can handle the axion electrodynamics in different states of
axions with different wavelength scales, not requiring axions to be at ground
state with a long wavelength. Thus, there is no need to neglect the spatial
gradient of the axion field ($\pmb{\nabla}a$ is finite) which is typically
assumed in the conventional approach. Our mathematical scheme aims to
providing a more precise and robust template for analyzing axion haloscopes in
several geometries of resonant cavities.

The paper is organized as follows: Sec. II presents the general solutions of
axion modified Maxwell equation in dual symmetric perspective and its
application to haloscope experiments; subsequently, some examples for
different types of cavity are presented in III, including observation of
resonance of axion-to-photon conversion power and the ratio of electric-
magnetic energy difference over the stored axion-induced electromagnetic
energy. Sec. IV presents some discussions between Sikivie and dual symmetric
axion electrodynamics theories while Sec. V devotes our conclusion.

##  II How haloscope works in dual symmetric point of view?

###  II.1 General solution of axion-induced electromagnetic field

To solve the $SO(2)$-invariant axion Maxwell equation [12, 13, 19, 20] in case
of no magnetic monopole:

$\displaystyle\left(\square+\dfrac{m_{a}^{2}c^{2}}{\hbar^{2}}\right)a$
$\displaystyle=$
$\displaystyle\dfrac{2g_{a\gamma\gamma}}{\mu_{0}c}\mathbf{E}\cdot\mathbf{B},$
(7)
$\displaystyle\pmb{\nabla}\cdot\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)=$
$\displaystyle\dfrac{\rho_{e}}{\varepsilon_{0}},$ (8a)
$\displaystyle\pmb{\nabla}\cdot\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)=$
$\displaystyle 0,$ (8b)
$\displaystyle\pmb{\nabla}\times\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)=$
$\displaystyle\dfrac{\partial\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)}{c\partial
t},$ (8c)
$\displaystyle\pmb{\nabla}\times\left(c\mathbf{B}+g_{a\gamma\gamma}a\mathbf{E}\right)=$
$\displaystyle\dfrac{\partial\left(\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B}\right)}{c\partial
t}+c\mu_{0}\mathbf{J}_{e},$ (8d)

we may set [12]

$\hat{\mathbf{E}}=\mathbf{E}-cg_{a\gamma\gamma}a\mathbf{B},\quad\hat{\mathbf{B}}=\mathbf{B}+\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E},$
(9)

i.e the electric and magnetic field strength $\mathbf{E}$, $\mathbf{B}$ can be
written in terms of $\hat{\mathbf{E}}$, $\hat{\mathbf{B}}$ as

$\displaystyle\mathbf{E}$ $\displaystyle=$
$\displaystyle\dfrac{\hat{\mathbf{E}}+cg_{a\gamma\gamma}a\hat{\mathbf{B}}}{1+g_{a\gamma\gamma}^{2}a^{2}},$
(10a) $\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\dfrac{\hat{\mathbf{B}}-g_{a\gamma\gamma}a\hat{\mathbf{E}}/c}{1+g_{a\gamma\gamma}^{2}a^{2}}.$
(10b)

Equations (8) in term of $\hat{\mathbf{E}}$, $\hat{\mathbf{B}}$ read

$\displaystyle\pmb{\nabla}\cdot\hat{\mathbf{E}}$ $\displaystyle=$
$\displaystyle\dfrac{\rho_{e}}{\varepsilon_{0}}$ (11a)
$\displaystyle\pmb{\nabla}\cdot\hat{\mathbf{B}}$ $\displaystyle=$
$\displaystyle 0$ (11b) $\displaystyle\pmb{\nabla}\times\hat{\mathbf{E}}$
$\displaystyle=$ $\displaystyle-\dfrac{\partial\hat{\mathbf{B}}}{\partial t}$
(11c) $\displaystyle\pmb{\nabla}\times\hat{\mathbf{B}}$ $\displaystyle=$
$\displaystyle\dfrac{1}{c^{2}}\dfrac{\partial\hat{\mathbf{E}}}{\partial
t}+\mu_{0}\mathbf{J}_{e},$ (11d)

which have the form of standard Maxwell equations without presence of axion
fields. Therefore, one may show that $\hat{\mathbf{E}}$ and $\hat{\mathbf{B}}$
are governed by the following D’Alembert equations [21]:

$\displaystyle\square\hat{\mathbf{E}}$ $\displaystyle=$
$\displaystyle-\dfrac{\pmb{\nabla}\rho_{e}}{\varepsilon_{0}}-\mu_{0}\dfrac{\partial\mathbf{J}_{e}}{\partial
t},$ (12a) $\displaystyle\square\hat{\mathbf{B}}$ $\displaystyle=$
$\displaystyle\mu_{0}\pmb{\nabla}\times\mathbf{J}_{e}.$ (12b)

Regarding the linearity of these above equations, their solutions can be
expressed as the linear combinations

$\hat{\mathbf{E}}=\mathbf{E}_{0}+\mathbf{E}_{rad},\quad\hat{\mathbf{B}}=\mathbf{B}_{0}+\mathbf{B}_{rad},$
(13)

between special solution $\mathbf{E}_{0}$, $\mathbf{B}_{0}$ which plays a role
as applied electromagnetic field and can be expressed by Jefimenko formula
(here $t_{r}=t-\left|\mathbf{r}-\mathbf{r}^{\prime}\right|/c$ is retarded
time) [21]

$\displaystyle\mathbf{E}_{0}\left(\mathbf{r},t\right)$
$\displaystyle=\dfrac{1}{4\pi\varepsilon_{0}}\int\left[\left(\dfrac{\rho_{e}(\mathbf{r}^{\prime},t_{r})}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}}+\dfrac{\partial_{t}\rho_{e}(\mathbf{r}^{\prime},t_{r})}{c\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{2}}\right)\left(\mathbf{r}-\mathbf{r}^{\prime}\right)-\dfrac{\partial_{t}\mathbf{J}_{e}(\mathbf{r}^{\prime},t_{r})}{c^{2}\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\right]d^{3}\mathbf{r}^{\prime},$
(14a) $\displaystyle\mathbf{B}_{0}\left(\mathbf{r},t\right)$
$\displaystyle=\dfrac{\mu_{0}}{4\pi}\int\left[\left(\dfrac{\mathbf{J}_{e}(\mathbf{r}^{\prime},t_{r})}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}}+\dfrac{\partial_{t}\mathbf{J}_{e}(\mathbf{r}^{\prime},t_{r})}{c\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{2}}\right)\times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\right]d^{3}\mathbf{r}^{\prime},$
(14b)

and the homogeneous solution $\mathbf{E}_{rad}$, $\mathbf{B}_{rad}$ which is
known as radiation electromagnetic field and can be presented in term of the
following Fourier integral [21]

$\displaystyle\mathbf{E}_{rad}(\mathbf{r},t)$
$\displaystyle=\int\pmb{\mathcal{E}}(\mathbf{r},\omega)e^{-\imath\omega
t}d\omega,$ (15a) $\displaystyle\mathbf{B}_{rad}(\mathbf{r},t)$
$\displaystyle=-\imath\int\pmb{\nabla}\times\pmb{\mathcal{E}}(\mathbf{r},\omega)e^{-\imath\omega
t}\omega^{-1}d\omega,$ (15b)

whereas Fourier image $\pmb{\mathcal{E}}(\mathbf{r},\omega)$ of free electric
component is governed by Helmholtz equation

$\left[\pmb{\nabla}\cdot\pmb{\nabla}+\omega^{2}/c^{2}\right]\pmb{\mathcal{E}}(\mathbf{r},\omega)=0.$
(16)

In principle, the solution of this equation can be expanded as linear
combination of its eigenfunction. However, the suitable eigenfunctions as well
as the coefficient of this expansion depend on the boundary condition of
electromagnetic field. Although we cannot provide the general expression of
free electromagnetic fields without knowing the exact boundary condition, the
magnitude of $\pmb{\mathcal{E}}(\mathbf{r},\omega)$ must be proportional to
axion-biphoton coupling constant $g_{a\gamma\gamma}$ due to the fact that
these radiation fields vanish when there is no axion $g_{a\gamma\gamma}=0$.

Since the coupling $g_{a\gamma\gamma}$ is weak, the Taylor expansion of
general solution of axion electromagnetic fields $\mathbf{E}$, $\mathbf{B}$ in
term of $g_{a\gamma\gamma}$ upto the first order is:

$\displaystyle\mathbf{E}$
$\displaystyle=\mathbf{E}_{0}+cg_{a\gamma\gamma}a\mathbf{B_{0}}+\mathbf{E}_{rad}+O(g^{2}_{a\gamma\gamma}),$
(17a) $\displaystyle\mathbf{B}$
$\displaystyle=\mathbf{B}_{0}-\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}+\mathbf{B}_{rad}+O(g_{a\gamma\gamma}^{2}).$
(17b)

In other words, the axion-induced electromagnetic fields are approximately

$\displaystyle\mathbf{E}_{ind}$ $\displaystyle\approx
cg_{a\gamma\gamma}a\mathbf{B}_{0}+\mathbf{E}_{rad},$ (18a)
$\displaystyle=cg_{a\gamma\gamma}a\mathbf{B}_{0}+\int\pmb{\mathcal{E}}(\mathbf{r},\omega)e^{-\imath\omega
t}d\omega,$ $\displaystyle\mathbf{B}_{ind}$
$\displaystyle\approx\mathbf{B}_{rad}-\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0},$
(18b)
$\displaystyle=-\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}-\imath\int\pmb{\nabla}\times\pmb{\mathcal{E}}(\mathbf{r},\omega)\frac{e^{-\imath\omega
t}}{\omega}d\omega.$

###  II.2 Axion haloscope under dual symmetric axion electrodynamics theory

For general haloscope experiments (see for examples [19, 18, 22, 23, 24] and
references therein), the induced electromagnetic fields $\mathbf{E}_{ind}$ and
$\mathbf{B}_{ind}$ are considered as response from the presence of axion when
nonzero external magnetic field $\mathbf{B}_{0}=\mathbf{B}_{0}(\mathbf{r})$
and zero external electric field i.e $\mathbf{E}_{0}=0$ are applied. The
Klein-Gordon equation of axion field (7) simply becomes the one of free axion:

$\left(\square+\dfrac{m_{a}^{2}c^{2}}{\hbar^{2}}\right)a=0.$ (19)

Spacetime evolution of axion field is a plane wave described as the following:

$a(\mathbf{r},t)=a_{0}(\omega)\exp\left[\imath\left(\mathbf{k}\cdot\mathbf{r}-\omega
t\right)\right],$ (20)

whereas $\omega=\sqrt{\mathbf{k}^{2}c^{2}+\omega_{a}^{2}}$ and
$\omega_{a}=m_{a}c^{2}/\hbar$.

Substituting $\mathbf{B}_{0}=\mathbf{B}_{0}(\mathbf{r})$, $\mathbf{E}_{0}=0$
and $a(\mathbf{r},t)$ from Equation (20) into Equation (18), we obtain the
axion-induced electromagnetic field

$\displaystyle\mathbf{E}_{ind}$
$\displaystyle=cg_{a\gamma\gamma}a(\mathbf{r},t)\mathbf{B}_{0}(\mathbf{r})+\int\pmb{\mathcal{E}}(\mathbf{r},\omega)e^{-\imath\omega
t}d\omega,$ (21a) $\displaystyle\mathbf{B}_{ind}$
$\displaystyle=-\imath\int\pmb{\nabla}\times\pmb{\mathcal{E}}(\mathbf{r},\omega)e^{-\imath\omega
t}\omega^{-1}d\omega.$ (21b)

To determine axion-induced electromagnetic field, we need to solve the Fourier
field $\pmb{\mathcal{E}}(\mathbf{r},\omega)$ via Helmholtz equation (16).
Denoting this Fourier field as
$\pmb{\mathcal{E}}(\mathbf{r},\omega)=-cg_{a\gamma\gamma}a(\mathbf{r},t)\pmb{\mathfrak{B}}(\mathbf{r},\omega)$,
the axion-induced electromagnetic field is expressed in a simpler form

$\displaystyle\mathbf{E}_{ind}(\mathbf{r},t)$
$\displaystyle=cg_{a\gamma\gamma}a_{0}(\omega)\left[\mathbf{B}_{0}(\mathbf{r})-\pmb{\mathfrak{B}}(\mathbf{r},\omega)\right]e^{\imath\left(\mathbf{k}(\omega)\cdot\mathbf{r}-\omega
t\right)},$ $\displaystyle\mathbf{B}_{ind}(\mathbf{r},t)$
$\displaystyle=\frac{\imath}{\omega}cg_{a\gamma\gamma}a_{0}(\omega)\left[\pmb{\nabla}\times\pmb{\mathfrak{B}}(\mathbf{r},\omega)\right.$
(22b)
$\displaystyle\quad\quad\quad\left.+\imath\mathbf{k}(\omega)\times\pmb{\mathfrak{B}}(\mathbf{r},\omega)\right]e^{\imath\left(\mathbf{k}(\omega)\cdot\mathbf{r}-\omega
t\right)}$

whereas $\pmb{\mathfrak{B}}(\mathbf{r},\omega)$ is governed by the following
Helmholtz equation:

$\left[\pmb{\nabla}\cdot\pmb{\nabla}+\omega^{2}/c^{2}\right]\pmb{\mathfrak{B}}(\mathbf{r},\omega)=0.$
(23)

The appearance of the term
$\imath\mathbf{k}(\omega)\times\pmb{\mathfrak{B}}(\mathbf{r},\omega)$ means
that the signal of axion-induced magnetic field depends on the direction of
axion motion. Suppose that axion moves in homogeneous space, it is necessary
to average the signal over all of possible direction of axion motion. While
$e^{\imath\mathbf{k}(\omega)\cdot\mathbf{r}}$ is nothing more than a phase
term of axion spacetime evolution which can be set up to unit by wisely
selected choice of initial phase as well as shifted origin of coordinate
system, the average of the term
$\imath\mathbf{k}(\omega)\times\pmb{\mathfrak{B}}(\mathbf{r},\omega)$ vanishes
because of the homogeneity assumption. Hence, the axion-induced
electromagnetic signal is simply

$\displaystyle\mathbf{E}_{ind}(\mathbf{r},t)=cg_{a\gamma\gamma}a_{0}\left(\omega\right)\left[\mathbf{B}_{0}(\mathbf{r})-\pmb{\mathfrak{B}}(\mathbf{r},\omega)\right]e^{-\imath\omega
t},$ (24a)
$\displaystyle\mathbf{B}_{ind}(\mathbf{r},t)=\frac{\imath}{\omega}cg_{a\gamma\gamma}a_{0}\left(\omega\right)\left[\pmb{\nabla}\times\pmb{\mathfrak{B}}(\mathbf{r},\omega)\right]e^{-\imath\omega
t}.$ (24b)

In Reference [25], cold axion or axion-like particles may form a Bose-Einstein
condensate, then the attitude of axion field of excited axion $\omega$ is much
smaller than that of ground state axion $\omega_{a}$ i.e
$|a_{0}(\omega)|\ll|a_{0}(\omega_{a})|$. Meanwhile, only the resonance at
ground state axion $\omega=\omega_{a}$ is considerable in haloscope
experiments. Consequently, all the problems in axion haloscopes are just to
find out the boundary condition for $\pmb{\mathfrak{B}}(\mathbf{r},\omega)$.
To illustrate this point, we consider the case of perfect electric conductor
resonant cavity as an example.

###  II.3 Solutions for perfect electric conductor resonant cavity

In many works of seeking axion signal, the perfect electric conductor (PEC) is
often used as resonant cavity surface to enhance the axion-induced
electromagnetic field. Since this surface, namely $C$, is perfectly
conductive, the total electric field must vanish in $C$ i.e
$\mathbf{E}(\mathbf{r}\in C,t)=0$. The DC-current is also prepared in this
resonant cavity surface i.e
$\mathbf{J}_{e}(\mathbf{r})=\mathbf{I}_{e}(\mathbf{r}_{C})\delta(\mathbf{r}\in
C)$. Then the boundary condition of the electric field
$\mathbf{E}(\mathbf{r}\in C,t)=0$ leads us to the boundary value problem of
$\pmb{\mathfrak{B}}(\mathbf{r},\omega)$

$\left\\{\begin{array}[]{c}\left[\pmb{\nabla}\cdot\pmb{\nabla}+\omega^{2}/c^{2}\right]\pmb{\mathfrak{B}}(\mathbf{r},\omega)=0\\\
\pmb{\mathfrak{B}}(\mathbf{r}_{C},\omega)=\mathbf{B}_{0}(\mathbf{r}_{C})\end{array}\right..$
(25)

In long-wavelength regime of axion field, most of axions lie in ground state
i.e $a_{0}(\omega)\approx a_{0}(\omega_{a})\delta(\omega-\omega_{a})$, then
the induced fields are simple harmonic oscillation at the frequency
$\omega=\omega_{a}$

$\displaystyle\mathbf{E}_{ind}^{LWA}(\mathbf{r},t)=$ $\displaystyle
cg_{a\gamma\gamma}a_{0}(\omega_{a})e^{-\imath\omega_{a}t}$ (26a)
$\displaystyle\times\left[\mathbf{B}_{0}(\mathbf{r})-\pmb{\mathfrak{B}}(\mathbf{r},\omega_{a})\right],$
$\displaystyle\mathbf{B}_{ind}^{LWA}(\mathbf{r},t)=$
$\displaystyle\frac{\imath}{\omega_{a}}cg_{a\gamma\gamma}a_{0}(\omega_{a})e^{-\imath\omega_{a}t}$
(26b)
$\displaystyle\times\left[\pmb{\nabla}\times\pmb{\mathfrak{B}}(\mathbf{r},\omega_{a})\right].$

In practice, suitable orthogonal coordinate system should be used to solve the
boundary value problem (25).

In PEC cavity, the electric and magnetic energy stored inside the cavity are
both conserved since they are just harmonic oscillating versus time. Thus the
summation and difference between electric and magnetic stored energy are also
conserved in this circumstance

$U_{\pm}(\omega)=\dfrac{\varepsilon_{0}}{4}\int_{C}\left(\mathbf{E}_{ind}\cdot\mathbf{E}_{ind}^{*}\pm
c^{2}\mathbf{B}_{ind}\cdot\mathbf{B}_{ind}^{*}\right)d^{3}\mathbf{r}.$ (27)

To characterize the stored energy induced from axion in PEC cavity, the
relevant quantities to examine are the form factor of the total energy
$C_{+}$, the energy difference $C_{-}$ and their ratio $\mathfrak{R}$:

$C_{\pm}(\omega)=\dfrac{U_{\pm}(\omega)}{\dfrac{g_{a\gamma\gamma}^{2}a_{0}^{2}}{2\mu_{0}}\int_{C}\mathbf{B}_{0}\cdot\mathbf{B}_{0}d^{3}\mathbf{r}},\quad\mathfrak{R}(\omega)=\left|\dfrac{C_{-}(\omega)}{C_{+}(\omega)}\right|$
(28)

It is observable that the form factor $C^{+}$ can be extremely large at some
certain frequencies $\omega_{c,n}$, namely cavity resonant frequencies. At
these resonant frequencies of the PEC cavity, the ratio $\mathfrak{R}$ also
vanishes. It is noticed that above results are also acceptable for imperfect
electric conductor if its thickness is such small that skin effect is
negligible.

###  II.4  The axion-to-photon conversion power in haloscope cavity

In realistic experiments, the haloscope cavity is not perfectly electric
conductive and the oscillation of induced electromagnetic field is damped i.e
the quality factor $Q_{c}$ of the cavity is finite. Additionally, the
oscillation of axion field is not simply harmonic one because of the thermal
distribution of its velocity under finite temperature above Bose-Einstein
condensate combining with effects from astrophysical motions [26, 27, 28].
Thus the attitude of axion field is approximately distributed as Cauchy-
Lorentz distribution around resonant at ground state axion $\omega_{a}$ with
the quality factor $Q_{a}$ which is around $10^{6}$ [28]. Taking into account
these factors, the axion-induced electromagnetic signal near resonant mode
$\omega_{c,n}$ is then modified as follows:

$\displaystyle\mathbf{E}_{ind}(\mathbf{r},t)$
$\displaystyle=cg_{a\gamma\gamma}a_{0}\left(\omega_{a}\right)\int\mathcal{A}(\omega,\omega_{a})\left[\mathbf{B}_{0}(\mathbf{r})-\mathcal{F}(\omega,\omega_{c,n})\pmb{\mathfrak{B}}(\mathbf{r},\omega)\right]e^{-\imath\omega
t}d\omega,$ $\displaystyle\mathbf{B}_{ind}(\mathbf{r},t)$
$\displaystyle=\imath\omega^{-1}cg_{a\gamma\gamma}a_{0}\left(\omega_{a}\right)\int\mathcal{A}(\omega,\omega_{a})\mathcal{F}(\omega,\omega_{c,n})\left[\pmb{\nabla}\times\pmb{\mathfrak{B}}(\mathbf{r},\omega)\right]e^{-\imath\omega
t}d\omega,$ (29a)

in which $\mathcal{A}(\omega,\omega_{a})$ and
$\mathcal{F}(\omega,\omega_{c,n})$ are two functions whose modules have form
of Cauchy-Lorentz distribution

$\displaystyle\mathcal{A}(\omega,\omega_{a})$
$\displaystyle=\sqrt{\dfrac{\omega_{a}}{2\pi
Q_{a}}}\dfrac{1}{\left(\omega-\omega_{a}\right)+\imath\frac{\omega_{a}}{2Q_{a}}},$
(30a) $\displaystyle\mathcal{F}(\omega,\omega_{c,n})$
$\displaystyle=\dfrac{1}{\sqrt{\pi}\left[\left(\omega-\omega_{c,n}\right)+\imath\frac{\omega_{c,n}}{2Q_{c}}\right]}.$
(30b)

Under this circumstance, the stored electric and magnetic field are not
conserved. The loss of stored energy corresponds to the conversion energy from
axion to photon: $a\to\gamma+\gamma$. Hence, the axion-to-photon conversion
power in the haloscope cavity $P_{a\to\gamma\gamma}(\omega_{a})$ is determined
as [23, 28]

$P_{a\to\gamma\gamma}(\omega_{a})=\dfrac{\omega_{c}}{Q_{c}}U_{tot}.$ (31)

with $U_{tot}$ is the average stored axion-induced electromagnetic energy
inside the cavity

$U_{tot}=\left\langle\dfrac{\varepsilon_{0}}{4}\int_{C}\left(\mathbf{E}_{ind}\cdot\mathbf{E}_{ind}^{*}\pm
c^{2}\mathbf{B}_{ind}\cdot\mathbf{B}_{ind}^{*}\right)d^{3}\mathbf{r}\right\rangle.$
(32)

Near the resonant mode $\omega_{a}\approx\omega_{c,n}$, the factor
$\left|\mathcal{F}(\omega,\omega_{c})\right|^{2}$ is much large than unit
while the ratio of the energy difference and total energy vanishes
$\mathfrak{R}\to 0$. As a result, the axion-to-photon conversion power is
approximately

$\displaystyle
P_{a\to\gamma\gamma}(\omega_{a})\approx\dfrac{g_{a\gamma\gamma}^{2}a_{0}^{2}}{\mu_{0}}\int_{C}\mathbf{B}_{0}\cdot\mathbf{B}_{0}d^{3}\mathbf{r}$
$\displaystyle\quad\times\frac{\omega_{c}}{2Q_{c}}\int\left|\mathcal{A}(\omega,\omega_{a})\right|^{2}\left|\mathcal{F}(\omega,\omega_{c})\right|^{2}C_{+}(\omega)d\omega$
(33)

Noticeably, when resonance occurs, $\omega_{a}\approx\omega_{c,n}$ are closed,
only the form factor $C_{+}(\omega)$ at $\omega=\omega_{a}$ contributes to the
axion-to-photon conversion power. In addition, the following integral becomes
Cauchy-Lorentz distribution:

$\displaystyle I(\omega_{a},\omega_{c,n})$
$\displaystyle=\dfrac{\omega_{c}}{2Q_{c}}\int\left|\mathcal{A}(\omega,\omega_{a})\right|^{2}\left|\mathcal{F}(\omega,\omega_{c})\right|^{2}d\omega$
(34)
$\displaystyle=\dfrac{\frac{\omega_{a}}{2Q}}{\pi\left[(\omega_{a}-\omega_{c,n})^{2}+\frac{\omega_{a}^{2}}{4Q^{2}}\right]}.$

whereas $Q=Q_{a}Q_{c}/(Q_{a}+Q_{c})$ is the effective quality factor which has
been recently established in Reference [28]. When both
$Q_{a},Q_{c}\to+\infty$, this integral coincides to delta Dirac distribution
$\delta(\omega_{a}-\omega_{c,n})$. This suggests that the finite quality
factors $Q_{a},Q_{c}$ move the pole of form factor $C_{+}(\omega)$ from
$\omega_{c,n}$ to $\omega_{c,n}\pm\imath\frac{\omega_{c,n}}{2Q}$. Hence, a
simple way to calculate the total stored energy $U_{tot}$ in imperfect
electric conductive cavity is to replace the form factor $C_{+}(\omega)$ of
PEC cavity by $\left|C_{+}\left(\omega(1+\imath 2^{-1}Q^{-1})\right)\right|$.
Finally, we obtain a simple approximation for axion-to-photon conversion power
in haloscope cavity

$\displaystyle
P_{a\to\gamma\gamma}(\omega_{a})\approx\dfrac{g_{a\gamma\gamma}^{2}a_{0}^{2}}{2\mu_{0}}\int_{C}\mathbf{B}_{0}\cdot\mathbf{B}_{0}d^{3}\mathbf{r}$
$\displaystyle\quad\quad\quad\quad\times\frac{\omega_{a}}{Q}\left|C_{+}\left(\omega_{a}\left(1+\frac{\imath}{2Q}\right)\right)\right|$
(35)

Hence, to detect the resonance of axion-to-photon conversion signal, we will
focus on the spectrum of the modified form factor
$Q^{-1}\left|C_{+}\left(\omega_{a}\left(1+\frac{\imath}{2Q}\right)\right)\right|$
versus axion angular frequency $\omega_{a}$. Moreover, we will observe the
zeros of the spectrum of the ratio $\mathfrak{R}(\omega_{a})$ in the PEC
cavity to point out the validity of the assumption $\mathfrak{R}\approx 0$
near resonance. We will consider the following cases: cylindrical solenoid,
spherical solenoid, two-parallel-sheet cavity, toroidal solenoid with a
rectangular cross-section and with a circular cross-section in the next
section.

##  III Resonance of axion-to-photon conversion power in different shapes of
cavity

We aim to consider the solution of axion electromagnetic field
$\mathbf{E}_{ind},\mathbf{B}_{ind}$ as well as the conversion power
$P_{a\to\gamma\gamma}(\omega_{a})$ and the radio $\mathfrak{R}(\omega_{a})$
for some different shapes of haloscope cavities such as a cylindrical
solenoid, spherical solenoid, two-parallel-sheet cavity, toroidal solenoid
with a rectangular cross-section and with a circular cross-section.

In our numerical estimation, we set up the size of the cavity for such the
resonance may occurs on the first mode i.e $\omega_{a}\approx\omega_{c,1}$.
According to many proposals such as ABRACADABRA [29], ADMX Collaboration [30,
31], HAYSTAC [32, 33, 34], ORGAN [35], BEAST [36] or MADMAX Collaboration [37,
22], the range of axion mass $m_{a}$ is around $0.1\mu eV$ to $100\mu eV$ and
hence, $1.5\times 10^{8}rad/s<\omega_{a}<1.5\times 10^{11}rad/s$. For
illustrative purpose, we will set first resonant frequency of the cavity
$\omega_{c,1}=1.5\times 10^{10}rad/s$. Meanwhile, our result is for searching
the axion like particle around $m_{a}\sim 10\mu eV$. Based on References [19,
23, 28], the quality factor of axion field $Q_{a}$ is around $1\times 10^{6}$
while the typical value of quality factor of the cavity is around $10^{4}$ to
$10^{6}$. Hence, we examine the effective $Q$ in the range from $10^{4}$ to
$10^{5}$.

Figure 1: Different shapes of cavity considered in this works: (a) cylindrical
solenoid, (b) spherical solenoid, (c) two-parallel-sheet cavity, (d) toroidal
solenoid with rectangular cross section and (e) with circular cross section.
The red arrow is the magnetic field and the blue arrow is the surface current.

###  III.1 Infinite cylindrical solenoid

In case of an infinite cylindrical solenoid whose radius is $R$, the surface
DC-current and its applied magnetic field are given in cylindrical coordinates
$(\rho,z,\phi)$ as

$\displaystyle\mathbf{J}_{e}(\mathbf{r})=\dfrac{B_{0}}{\mu_{0}}\delta(\rho-R)\pmb{\hat{\phi}},\quad\mathbf{B}_{0}(\mathbf{r})=B_{0}\theta(R-\rho)\hat{\mathbf{z}},$
(36)

with $\delta$ and $\theta$ are notations of Dirac delta and Heaviside theta
functions. Replacing $\mathbf{B}_{0}$ of Equation (36) into Equation (25) and
solving this boundary value problem in cylindrical coordinates $(\rho,z,\phi)$
give us

$\displaystyle\pmb{\mathfrak{B}}(\rho,\omega)=B_{0}\dfrac{J_{0}(\rho\omega/c)}{J_{0}(R\omega/c)}\hat{\mathbf{z}},$
(37a)

with $J_{n}(x)$ denotes for Bessel J function. Consequently, the axion-induced
fields are explicitly determined

$\displaystyle\mathbf{E}_{ind}(\rho,t)$
$\displaystyle=cg_{a\gamma\gamma}a_{0}B_{0}e^{-\imath\omega_{a}t}$ (38a)
$\displaystyle\times\left[\theta(R-\rho)-\dfrac{J_{0}(\rho\omega_{a}/c)}{J_{0}(R\omega_{a}/c)}\right]\hat{\mathbf{z}},$
$\displaystyle\mathbf{B}_{ind}(\rho,t)$ $\displaystyle=\imath
g_{a\gamma\gamma}a_{0}B_{0}\dfrac{J_{1}(\rho\omega_{a}/c)}{J_{0}(R\omega_{a}/c)}e^{-\imath\omega_{a}t}\pmb{\hat{\phi}},$
(38b)

as the same as results in Reference [23]. Since the explicit expression of
axion-induced electromagnetic field is known, it is easy to determine both
total/difference EM form factor $C_{\pm}$ and their ratio $\mathfrak{R}$

$\displaystyle C_{+}$
$\displaystyle=\dfrac{J_{1}^{2}(R\omega_{a}/c)}{J_{0}^{2}(R\omega_{a}/c)}-\dfrac{3J_{2}(R\omega_{a}/c)}{2J_{0}(R\omega_{a}/c)}$
(39a) $\displaystyle C_{-}$
$\displaystyle=-\dfrac{J_{2}(R\omega_{a}/c)}{2J_{0}(R\omega_{a}/c)}$ (39b)
$\displaystyle\mathfrak{R}$
$\displaystyle=\dfrac{1}{\left|3-\dfrac{2J_{1}^{2}(R\omega_{a}/c)}{J_{0}(R\omega_{a}/c)J_{2}(R\omega_{a}/c)}\right|}.$
(39c)

As can be seen from Equation (39a), the form factor $C_{+}(\omega_{a})$ is
resonant at $\omega_{c,n}=\chi_{n}c/R$, whereas $\chi_{n}$ is $n^{th}$-zero of
$J_{0}(\chi)$ function e.g
$\chi_{n}=2.40483,\;5.52008,\;8.65373,\;11.7915,\;14.9309\ldots$ for
$n=1,2,3,4,5\ldots$.

We choose the solenoid radius $R=4.80965\text{ cm}$ so that the axion signal
is resonant at $m_{a}\sim 10\mu eV$. Figure 2 shows axion-induced
electromagnetic fields strength $E_{ind},B_{ind}$ near first three resonant
modes varying in space while Figure 14 indicates the distribution of the
axion-induced electromagnetic stored energy density. Figure 3 illustrates the
modified form factor of imperfect electric conductive cavity
$\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus $\omega_{a}/\omega_{c,1}$.

Figure 2: (a,b,c) The axion-induced electromagnetic fields strength
$E_{ind},B_{ind}$ near first three resonant modes
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$ vary in space
respectively.

Figure 3: Profile of the modified form factor of imperfect electric conductive
cavity $\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus frequency ratio $\omega_{a}/\omega_{c,1}$
for infinite cylindrical solenoid cavity. Black, blue and red curves are for
effective $Q$-factor are $10^{4}$, $5\times 10^{4}$, $10^{5}$ respectively.
The higher quality factor is, the sharper resonant peaks are.

###  III.2 Spherical solenoid

If the resonant cavity is a spherical superconductor whose radius is $R$, the
spherical coordinates $(r,\theta,\varphi)$ should be used. The spherical
surface DC-current and its corresponding magnetic field _inside the cavity_
are:

$\displaystyle\left\\{\begin{array}[]{l}\mathbf{J}_{e}(\mathbf{r})=\dfrac{3B_{0}\sin\theta}{2\mu_{0}}\delta(r-R)\left[\hat{\mathbf{y}}\cos\varphi-\hat{\mathbf{x}}\sin\varphi\right],\\\
\mathbf{B}_{0}(\mathbf{r})=B_{0}\hat{\mathbf{z}}\text{ for
}r<R.\end{array}\right..$ (42)

According to the procedure in Equation (25) and Equation (26), we obtain
inside the cavity $r<R$ that

$\displaystyle\pmb{\mathfrak{B}}(r,\omega)$
$\displaystyle=B_{0}\dfrac{R}{r}\dfrac{\sin\left(r\omega/c\right)}{\sin\left(R\omega/c\right)}\hat{\mathbf{z}},$
(43a) $\displaystyle\mathbf{E}_{ind}(r,t)$
$\displaystyle=cg_{a\gamma\gamma}a_{0}B_{0}\left[1-\dfrac{R}{r}\dfrac{\sin\left(r\omega/c\right)}{\sin\left(R\omega/c\right)}\right]e^{-\imath\omega_{a}t},$
(43b) $\displaystyle\mathbf{B}_{ind}(r,t)$ $\displaystyle=\imath
g_{a\gamma\gamma}a_{0}B_{0}\dfrac{R}{r}\dfrac{\sin\left(r\omega/c\right)}{\sin\left(R\omega/c\right)}e^{-\imath\omega_{a}t}\times$
(43c)
$\displaystyle\times\left[\dfrac{c}{r\omega_{a}}-\cot\left(\dfrac{r\omega_{a}}{c}\right)\right]\left[\hat{\mathbf{y}}\cos\varphi-\hat{\mathbf{x}}\sin\varphi\right].$

It is noticed that in this case, the applied field as well as the axion-
induced field do not vanish outside the cavity. Only the the one stored inside
the cavity is significantly considered. Hence, the total/difference EM form
factor and their ratio inside the cavity are

$\displaystyle C_{+}$
$\displaystyle=2+\dfrac{3}{2}\left[\cot\left(\dfrac{R\omega_{a}}{c}\right)+\dfrac{c}{R\omega_{a}}\right]^{2}-\dfrac{6c^{2}}{R^{2}\omega_{a}^{2}}$
(44a) $\displaystyle C_{-}$
$\displaystyle=\dfrac{1}{2}+\dfrac{3c}{2R\omega_{a}}\left(\cot\left(\dfrac{R\omega_{a}}{c}\right)-\dfrac{c}{R\omega_{a}}\right)$
(44b) $\displaystyle\mathfrak{R}$
$\displaystyle=\left|\dfrac{1}{2}+\dfrac{3c}{2R\omega_{a}}\left(\cot\left(\dfrac{R\omega_{a}}{c}\right)-\dfrac{c}{R\omega_{a}}\right)\right|\times$
(44c)
$\displaystyle\times\left|2+\dfrac{3}{2}\left[\cot\left(\dfrac{R\omega_{a}}{c}\right)+\dfrac{c}{R\omega_{a}}\right]^{2}-\dfrac{6c^{2}}{R^{2}\omega_{a}^{2}}\right|^{-1}.$

The resonance of the axion-photon signal occurs when $cotan$ in the form
factor $C_{+}$ is infinity, it means that the axion frequency approaches
$\omega_{c,n}=n\pi c/R$ for $n=1,2,3,\ldots$.

To enhance the axion signal at $m_{a}\sim 10\mu eV$, we set the radius
$R=6.28319\text{ cm}$. The profile of the modified form factor of imperfect
electric conductive cavity $\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus $\omega_{a}/\omega_{c,1}$ are shown in
Figure 4.

Figure 4: Profile of the modified form factor of imperfect electric conductive
cavity $\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus frequency ratio $\omega_{a}/\omega_{c,1}$
for spherical cavity. Black, blue and red curves are for effective $Q$-factor
are $10^{4}$, $5\times 10^{4}$, $10^{5}$ respectively. The higher quality
factor is, the sharper resonant peaks are.

Figure 5 shows axion-induced electromagnetic fields strength $E_{ind},B_{ind}$
near first three resonant modes varying in space while Figure 15 indicates the
distribution of the axion-induced electromagnetic stored energy density at the
equator of the spherical cavity.

Figure 5: (a,b,c) The axion-induced electromagnetic fields strength
$E_{ind},B_{ind}$ near first three resonant modes
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$ vary in space
respectively.

###  III.3 Infinite two-parallel-sheet cavity

The infinite two-parallel-sheet cavity consists two infinite plates which are
placed parallel to each other and distances by $2R$. In this cavity, the
surface DC-current and its applied magnetic field are given in Cartesian
coordinates $(x,y,z)$ as

$\displaystyle\left\\{\begin{array}[]{l}\mathbf{J}_{e}(\mathbf{r})=\dfrac{2B_{0}}{\mu_{0}}\left[\delta(x-R)-\delta(x+R)\right]\hat{\mathbf{y}},\\\
\mathbf{B}_{0}(\mathbf{r})=B_{0}\left[\theta\left(x-R\right)-\theta\left(x+R\right)\right]\hat{\mathbf{z}}.\end{array}\right..$
(47)

Replacing $\mathbf{B}_{0}$ of Equation (47) into Equation (25) followed by
solving this boundary value problem in Cartesian coordinates $(x,y,z)$ leads
us to the exact expression of axion-induced electric and magnetic fields
inside the cavity $|x|<R$

$\displaystyle\pmb{\mathfrak{B}}(\rho,\omega)$
$\displaystyle=B_{0}\dfrac{\cos(x\omega/c)}{\cos(R\omega/c)}\hat{\mathbf{z}},$
(48a) $\displaystyle\mathbf{E}_{ind}(\rho,t)$
$\displaystyle=cg_{a\gamma\gamma}a_{0}B_{0}\left[1-\dfrac{\cos(x\omega/c)}{\cos(R\omega/c)}\right]e^{-\imath\omega_{a}t}\hat{\mathbf{z}}.$
(48b) $\displaystyle\mathbf{B}_{ind}(\rho,t)$ $\displaystyle=\imath
g_{a\gamma\gamma}a_{0}B_{0}\dfrac{\sin(x\omega/c)}{\cos(R\omega/c)}e^{-\imath\omega_{a}t}\hat{\mathbf{y}},$
(48c)

As long as the explicit expression of axion-induced electromagnetic field is
known, both total/difference EM form factor $C_{\pm}$ and their ratio
$\mathfrak{R}$ are known

$\displaystyle C_{+}$
$\displaystyle=1-\dfrac{\tan\left(R\omega_{a}/c\right)}{R\omega_{a}/c}+\dfrac{\tan^{2}\left(R\omega_{a}/c\right)}{2}$
(49a) $\displaystyle C_{-}$
$\displaystyle=\dfrac{1}{2}\left[1-\dfrac{\tan\left(R\omega_{a}/c\right)}{R\omega_{a}/c}\right]$
(49b) $\displaystyle\mathfrak{R}$
$\displaystyle=\dfrac{1}{2}\left|\dfrac{1-\dfrac{\tan\left(R\omega_{a}/c\right)}{R\omega_{a}/c}}{1-\dfrac{\tan\left(R\omega_{a}/c\right)}{R\omega_{a}/c}+\dfrac{\tan^{2}\left(R\omega_{a}/c\right)}{2}}\right|.$
(49c)

When $\omega_{a}\to\omega_{c,n}=(n-1/2)\pi c/R\;(n=1,2,\ldots)$, the form
factor $C_{+}$ goes to infinity and the ratio $\mathfrak{R}$ vanishes in PEC
cavity. Therefore, the axion-to-photon conversion power for imperfect electric
conductive cavity with the same shape is resonant at $\omega_{c,n}=(n-1/2)\pi
c/R$.

The gap between the sheets is set at $R=3.141595\text{ cm}$ so that the axion
signal is resonant at $m_{a}\sim 10\mu eV$. Figure 6 shows axion-induced
electromagnetic fields strength $E_{ind},B_{ind}$ near first three resonant
modes varying in space while Figure 16 indicates the distribution of the
axion-induced electromagnetic stored energy density. Figure 7 shows the
dependence of the modified form factor of imperfect electric conductive cavity
$\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ and the ratio $\mathfrak{R}(\omega_{a})$ on
$\omega_{a}/\omega_{c,1}$.

Figure 6: (a,b,c) The axion-induced electromagnetic fields strength
$E_{ind},B_{ind}$ near first three resonant modes
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$ vary in space
respectively.

Figure 7: Profile of the modified form factor of imperfect electric conductive
cavity $\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus frequency ratio $\omega_{a}/\omega_{c,1}$
for two-parallel-sheet cavity. Black, blue and red curves are for effective
$Q$-factor are $10^{4}$, $5\times 10^{4}$, $10^{5}$ respectively.

###  III.4 Toroidal solenoid with rectangular cross section

Considering the rectangular cross-section toroidal cavity with half width $R$,
height $h$ and average radius $D$, the cavity surfaces in cylindrical
coordinate are $\rho=D\pm R$ and $z=\pm h/2$. The applied magnetic field
exists only inside the cavity

$\mathbf{B}_{0}\left(\mathbf{r}\right)=\dfrac{B_{0}D}{\rho}\left[\hat{\mathbf{y}}\cos\phi-\hat{\mathbf{x}}\sin\phi\right]$
(50)

when $|\rho-D|\leq R$ and $|z|\leq h/2$. Solving the boundary value problem
(25) inside the cavity gives

$\displaystyle\pmb{\mathfrak{B}}(\rho,\omega)$
$\displaystyle=B_{0}\left[\alpha J_{1}\left(\rho\omega/c\right)+\beta
Y_{1}\left(\rho\omega/c\right)\right]\left[\hat{\mathbf{y}}\cos\phi-\hat{\mathbf{x}}\sin\phi\right],$
(51a) $\displaystyle\mathbf{E}_{ind}(\rho,\omega)$
$\displaystyle=cg_{a\gamma\gamma}a_{0}B_{0}\left[\dfrac{D}{\rho}-\alpha
J_{1}(\rho\omega/c)-\beta Y_{1}(\rho\omega/c)\right]e^{-\imath\omega
t}\left[\hat{\mathbf{y}}\cos\phi-\hat{\mathbf{x}}\sin\phi\right]$ (51b)
$\displaystyle\mathbf{B}_{ind}(\rho,\omega)$ $\displaystyle=\imath
g_{a\gamma\gamma}a_{0}B_{0}\left[\alpha J_{0}(\rho\omega/c)+\beta
Y_{0}(\rho\omega/c)\right]e^{-\imath\omega t}\hat{\mathbf{z}}$ (51c)

in which

$\displaystyle\begin{pmatrix}\alpha\\\ \beta\end{pmatrix}$
$\displaystyle=\dfrac{D/(D^{2}-R^{2})}{J_{1}((D-R)\omega/c)Y_{1}((D+R)\omega/c)-J_{1}((D+R)\omega/c)Y_{1}((D-R)\omega/c)}\times$
$\displaystyle\begin{pmatrix}(D+R)Y_{1}((D+R)\omega/c)-(D-R)Y_{1}((D-R)\omega/c)\\\
(D-R)J_{1}((D-R)\omega/c)-(D+R)J_{1}((D+R)\omega/c)\end{pmatrix},\quad\quad
Y_{n}(x)\text{ is Bessel Y function.}$

This result coincides to the one in Reference [23]. The form factors $C_{\pm}$
as well as their ratio $\mathfrak{R}$ are acquired using some integrals of
Bessel function provided in [38]

$\displaystyle
C_{\pm}=1+\left[\frac{2\omega_{a}^{2}D^{2}}{c^{2}}\ln\left(\frac{D+R}{D-R}\right)\right]^{-1}\biggl{\\{}\frac{\alpha^{2}z^{2}}{2}\left(J_{1}^{2}(z)-J_{2}(z)J_{0}(z)\pm(J_{1}^{2}(z)+J_{0}^{2}(z))\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\biggl{.}+\frac{\beta^{2}z^{2}}{2}\left(Y_{1}^{2}(z)-Y_{2}(z)Y_{0}(z)\pm(Y_{1}^{2}(z)+Y_{0}^{2}(z))\right)-\frac{2\omega_{a}D}{c}\left(\alpha
J_{0}(z)+\beta Y_{0}(z)\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\biggl{.}+\alpha\beta
z^{2}\left(J_{1}(z)Y_{1}(z)-\frac{J_{0}(z)Y_{2}(z)+J_{2}(z)Y_{0}(z)}{2}\pm(J_{0}(z)Y_{0}(z)+J_{1}(z)Y_{1}(z))\right)\biggr{\\}}_{\omega_{a}(D-R)/c}^{\omega_{a}(D+R)/c},$
(52a) $\displaystyle\mathfrak{R}=\left|\dfrac{C_{-}}{C_{+}}\right|.$ (52b)

About resonant angular frequencies $\omega_{c,n}$, we look on the nodes of the
denominator of $\alpha$ and $\beta$ coefficient i.e

$\begin{array}[]{ll}f(D,R,\omega)=&J_{1}((D-R)\omega/c)Y_{1}((D+R)\omega/c)\\\
&-J_{1}((D+R)\omega/c)Y_{1}((D-R)\omega/c),\end{array}$

and the nodes of their numerators $g_{\alpha}(D,R,\omega)=\alpha
f(D,R,\omega),g_{\beta}(D,R,\omega)=\beta f(D,R,\omega)$. When the average
radius of the toroidal solenoid is large $D\to\infty$, its cylinders becomes
flat sheets distanced by $2R$. This suggests the resonance of the toroidal
solenoid should occurs near the resonance of two-parallel-sheet cavity gapped
by $2R$. Meanwhile, $\lim\limits_{D/R\to\infty}\omega_{c,n}=(n-1/2)\pi c/R$
and the approximation $\omega_{c,n}\approx(n-1/2)\pi c/R$ is meaningful when
the average radius of the rectangular-torus cavity is larger than a certain
value. Observation on the nodes of $f(D,R,\omega)$ and
$g_{\alpha,\beta}(D,R,\omega)$, which is illustrated in Figure 8, indicates
that zeros of $f(D,R,\omega)$ tend to $(n-1/2)\pi c/R$ or $n\pi c/R$ for
$D>2R$ while zeros of $g_{\alpha,\beta}(D,R,\omega)$ tend to $n\pi c/R$ for
$D>2R$. The resonant angular frequencies $\omega_{c,n}$ must be the
frequencies at which the denominator $f(D,R,\omega)$ vanishes while the
numerators $g_{\alpha,\beta}(D,R,\omega)$ do not. So the approximation
$\omega_{c,n}\approx(n-1/2)\pi c/R$ is hold for $D>2R$.

Figure 8: Zeros of (a) denominator $f(D,R,\omega)$ and (b,c) numerators
$g_{\alpha,\beta}(D,R,\omega)$ versus $D/R$ when $D\geq 2R$. Since the zeros
of denominator $f(D,R,\omega)$ and (b,c) numerators
$g_{\alpha,\beta}(D,R,\omega)$ coincide near $\omega\approx n\pi c/R$, the
possible resonant frequencies are $(n-1/2)\pi c/R$. (d,e) Profile of
coefficients $\alpha$ (blue, thick), $\beta$ (red, dashed) versus $\omega R/c$
when $D=2R$ and $D=5R$ respectively. $\alpha$ and $\beta$ go to very large
value when $\omega$ approach $\omega_{c,n}\approx(n-1/2)\pi c/R$.

Similar to the previous cases, we choose the half distance between two
cylinders of the rectangular-torus cavity $R=3.141595\text{ cm}$ so that the
axion signal is approximately resonant at $m_{a}\sim 10\mu eV$. The average
radius is chosen as $D=5R$ i.e the outer cylinder is $1.5$ times bigger than
the inner one. Figure 9 shows the dependence of the modified form factor of
imperfect electric conductive cavity
$\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ and the ratio $\mathfrak{R}(\omega_{a})$ on
$\omega_{a}/\omega_{c,1}$.

Figure 9: Profile of the modified form factor of imperfect electric conductive
cavity $\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus frequency ratio $\omega_{a}/\omega_{c,1}$
for toroidal solenoid with a rectangular cross-section in case $D=5R$. Black,
blue and red curves are for effective $Q$-factor are $10^{4}$, $5\times
10^{4}$, $10^{5}$ respectively.

When looking at the Figure 9 for the first moment, one may think that there is
resonant at $\omega_{a}=2\omega_{c,1}$. However, this is the false resonant
arising from numerical error of the estimation $\omega_{c,n}\approx(n-1/2)\pi
c/R$. When increasing the ratio $D/R$ the signal at this point goes down
significantly and the profile will look like the one of two-parallel-sheet
cavity when $D$ is large enough. For example, Figure 10 shows the profile when
$D=100R$.

Figure 10: Profile of the modified form factor of imperfect electric
conductive cavity $\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus frequency ratio $\omega_{a}/\omega_{c,1}$
for toroidal solenoid with a rectangular cross-section in case $D=100R$.
Black, blue and red curves are for effective $Q$-factor are $10^{4}$, $5\times
10^{4}$, $10^{5}$ respectively.

Figure 11 shows axion-induced electromagnetic fields strength
$E_{ind},B_{ind}$ near first three resonant modes varying in space while
Figure 17 indicates the distribution of the axion-induced electromagnetic
stored energy density.

Figure 11: (a,b,c) The axion-induced electromagnetic fields strength
$E_{ind},B_{ind}$ near first three resonant modes
$\omega_{a}=1.1\omega_{c,1},1.1\omega_{c,2},1.1\omega_{c,3}$ vary in space
respectively.

###  III.5 Toroidal solenoid with circular cross section

One of the most attention of theoretical investigation on cold axion searches
is about using circular cross-section toroidal solenoid as a resonant cavity
because this cavity is more practical than infinite cylindrical solenoid and
its magnetic field is nearly uniform. Different from previous cases, the set
of orthogonal coordinates namely toroidal coordinates $(\xi,\eta,\varphi)$
[39]

$\left\\{\begin{array}[]{l}x=\dfrac{d\sinh\xi\cos\varphi}{\cosh\xi-\cos\eta},\\\
y=\dfrac{d\sinh\xi\sin\varphi}{\cosh\xi-\cos\eta},\\\
z=\dfrac{d\sin\eta}{\cosh\xi-\cos\eta}.\end{array}\right.$

In this paper, we use a different coordinate system from other works such as
in [19, 23] because the one used here is orthogonal and Helmholtz equation has
been solved in this kind of coordinate system [40]. The surface of a torus
with major radius $D$ and minor radius $R$ is simply described as a surface
$\xi=\ln\left[\left(D+d\right)/2R\right]$ with $d=\sqrt{D^{2}-R^{2}}$. As
Laplace equation is separably solvable in this coordinate system, the applied
magnetic field inside this toroidal cavity can be analytically expressed by a
well-know form

$\displaystyle\mathbf{B}_{0}(\mathbf{r})$
$\displaystyle=\dfrac{B_{0}D}{\sqrt{x^{2}+y^{2}}}\left[\hat{\mathbf{y}}\cos\varphi-\hat{\mathbf{x}}\sin\varphi\right]$
(53a)
$\displaystyle=\dfrac{B_{0}D\left(\cosh\xi-\cos\eta\right)}{d\sinh\xi}\left[\hat{\mathbf{y}}\cos\varphi-\hat{\mathbf{x}}\sin\varphi\right]$

Regretfully, the Helmholtz equation (25) is inseparable in toroidal
coordinetes so that the boundary value problem (25) must be solved by
recurrence method [40]. Right at the surface inside the cavity
$\xi\to\xi_{0}=\ln\left[\left(D+d\right)/2R\right]^{-}$, three components of
applied magnetic field are
$\left(B_{0,x},B_{0,y},B_{0,z}\right)=B_{0}Dd^{-2}\left(D-R\cos\eta\right)\times\left(-\sin\varphi,\cos\varphi,0\right)$.
Therefore, the solution of Equation (25) should take the form

$\pmb{\mathfrak{B}}(\mathbf{r},\omega)=B_{0}F\left(\xi,\eta,\omega\right)\left[\hat{\mathbf{y}}\cos\varphi-\hat{\mathbf{x}}\sin\varphi\right],$

where $F\left(\xi,\eta,\omega\right)=\sum_{n}c_{n}W_{n}(\xi,\eta,\omega)$ is
linear combination of non-separated eigenfunctions $W_{n}(\xi,\eta,\omega)$
with the coefficients $c_{n}$ satisfying
$F\left(\xi_{0},\eta,\omega\right)=Dd^{-2}\left(D-R\cos\eta\right)$. Reference
[40] showed that non-separated eigenfunctions $W_{n}(\xi,\eta,\omega)$ had to
take the following forms

$\displaystyle W_{n}(\xi,\eta,\omega)$
$\displaystyle=\sum_{r=T}^{\infty}A_{r}(\xi)\left(\dfrac{\cosh\xi-\cos\eta}{d\sinh\xi}\right)^{-r},$
$\displaystyle W_{0}(\xi,\eta,\omega)$
$\displaystyle=\sin\eta\sum_{r=T^{\prime}}^{\infty}B_{r}(\xi)\left(\dfrac{\cosh\xi-\cos\eta}{d\sinh\xi}\right)^{-r},$

whereas $A_{r}(s)$ and $B_{r}(s)$ are solutions of a set of recurrence
differential equations. Comparing with the boundary condition of
$F\left(\xi,\eta,\omega\right)$, its general solution should be

$\displaystyle F(\xi,\eta,\omega)$
$\displaystyle=Df_{-1}(\xi,\omega)\left(\dfrac{\cosh\xi-\cos\eta}{d\sinh\xi}\right)+$
$\displaystyle\sum_{r=0}^{\infty}D^{-r}f_{r}(\xi,\omega)\left(\dfrac{\cosh\xi-\cos\eta}{d\sinh\xi}\right)^{-r}.$

To avoid cumbersome arising from full solution of
$F\left(\xi,\eta,\omega\right)$, we perform the approximate solution by two
following assumption: (i) the major radius $D$ is much larger than minor
radius $R$, (ii) axion frequency $\omega$ is quite small enough. The former
assumption comes from the asymptotic behaviour that the circular toroidal
cavity coincides the circular cylindrical solenoid when $D/R\to\infty$. The
second assumption is based on the other asymptotic behaviour that
$F\left(\xi,\eta,\omega\right)$ becomes $|\mathbf{B}_{0}(\xi,\eta)|/B_{0}$
when $\omega\to 0$. As the results, we neglect all of higher order terms
$r\geq 0$ i.e $f_{r}(\xi,\omega)\approx 0$. The suitable approximate solution
is given as

$F(\xi,\eta,\omega)\approx
Df(\xi,\omega)\left(\dfrac{\cosh\xi-\cos\eta}{d\sinh\xi}\right).$

Introducing new variable $z=\omega dc^{-1}\cosh^{-1}\xi\ll 1$ and neglecting
$\eta$ terms as well as higher order terms of $z$, we show that $f(z)$ obeys
the approximate ordinary differential equation

$z^{2}f^{{}^{\prime\prime}}(z)+zf^{{}^{\prime}}(z)+z^{2}f(s)\approx 0,$

whose solution is the Bessel $J$ function: $f(z)\sim J_{0}(z).$ [38]. Finally,
we obtain 111It is noticed that if $\rho$ is distance from a point to the
circle $r=D$ in $Oxy$ plane and the major radius $D$ is much larger than minor
radius $R$, then $\rho\approx d/\cosh\xi$ and hence, above results coincides
to the known one in References [23, 19]
$\pmb{\mathfrak{B}}(\mathbf{r},\omega)\approx\mathbf{B}_{0}\dfrac{J_{0}(\rho\omega/c)}{J_{0}(R\omega/c)}$.

$\displaystyle\pmb{\mathfrak{B}}(\mathbf{r},\omega)\approx\mathbf{B}_{0}(\mathbf{r})\dfrac{J_{0}\left(\dfrac{\omega\sqrt{D^{2}-R^{2}}}{c}\dfrac{1}{\cosh\xi}\right)}{J_{0}\left(\dfrac{\omega
R}{c}\dfrac{\sqrt{D^{2}-R^{2}}}{D}\right)},$ (54)

from which the axion-induced fields, the total and difference EM form factors
as well as their ratio are calculated

$\displaystyle\mathbf{E}_{ind}(\mathbf{r},\omega)\approx
cg_{a\gamma\gamma}a_{0}B_{0}e^{-\imath\omega
t}\left[\hat{\mathbf{y}}\cos\varphi-\hat{\mathbf{x}}\sin\varphi\right]\dfrac{D(\cosh\xi-\cos\eta)}{d\sinh\xi}\left[1-\dfrac{J_{0}\left(\frac{\omega\sqrt{D^{2}-R^{2}}}{c}\frac{1}{\cosh\xi}\right)}{J_{0}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}\right].$ (55a)
$\displaystyle\mathbf{B}_{ind}(\mathbf{r},\omega)\approx\imath
g_{a\gamma\gamma}a_{0}B_{0}e^{-\imath\omega
t}\frac{D\left(\cosh\xi-\cos\eta\right)^{2}}{d\cosh^{2}\xi}\dfrac{J_{1}\left(\frac{\omega\sqrt{D^{2}-R^{2}}}{c}\frac{1}{\cosh\xi}\right)}{J_{0}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}$
$\displaystyle\times\left[\frac{\cosh\xi\sin\eta}{\cosh\xi-\cos\eta}\left(\hat{\mathbf{x}}\cos\varphi+\hat{\mathbf{y}}\sin\varphi\right)+\frac{1-\cosh\xi\cos\eta}{\cosh\xi-\cos\eta}\hat{\mathbf{z}}\right],$
(55b) $\displaystyle C_{\pm}\approx 1-\dfrac{2}{\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}}\dfrac{J_{1}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}{J_{0}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}+\frac{1}{2}\dfrac{J_{1}^{2}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}{J_{0}^{2}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}\pm\frac{1}{2}\dfrac{J_{1}^{2}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}{J_{0}^{2}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}\mp\frac{1}{2}\dfrac{J_{2}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}{J_{0}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}$ (55c)
$\displaystyle\mathfrak{R}=\dfrac{1}{\left|3-\frac{2J_{1}^{2}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}{J_{0}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)J_{2}\left(\frac{\omega
R}{c}\frac{\sqrt{D^{2}-R^{2}}}{D}\right)}\right|}.$ (55d)

As can be seen from above expression, the results in circular toroidal cavity
is similar to the one in infinite cylindrical cavity whereas the frequency is
scaled by a factor $\sqrt{D^{2}-R^{2}}/D$ which is closed to $1$. Therefore,
these results obviously take infinite cylindrical cavity as its limit. In the
same manner as infinite cylindrical cavity, the resonant frequency of circular
toroidal cavity is given as $\omega_{c,n}=\chi_{n}cD/R\sqrt{D^{2}-R^{2}}$ in
which $\chi_{n}$ is $n^{th}$-zero of $J_{0}(\chi)$ function.

Similar to infinite cylindrical cavity, we choose the minor radius
$R=4.90883\text{ cm}$ while the major radius is 5 times of that $D=5R$. As the
result, the axion signal is resonant at $m_{a}\sim 10\mu eV$. Figure 12
illustrates the modified form factor of imperfect electric conductive cavity
$\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus $\omega_{a}/\omega_{c,1}$.

Figure 12: Profile of the modified form factor of imperfect electric
conductive cavity $\left|Q^{-1}C_{+}\left(\omega_{a}\left(1+\imath
2^{-1}Q^{-1}\right)\right)\right|$ as well as the ratio
$\mathfrak{R}(\omega_{a})$ versus frequency ratio $\omega_{a}/\omega_{c,1}$
for circular toroidal solenoid cavity. Black, blue and red curves are for
effective $Q$-factor are $10^{4}$, $5\times 10^{4}$, $10^{5}$ respectively.
The higher quality factor is, the sharper resonant peaks are.

Figure 13 shows axion-induced electromagnetic fields strength
$E_{ind},B_{ind}$ near first three resonant modes varying in space while
Figure 18 indicates the distribution of the axion-induced electromagnetic
stored energy density.

Figure 13: (a,b,c) The axion-induced electromagnetic fields strength
$E_{ind},B_{ind}$ near first three resonant modes
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$ vary in space
respectively.

##  IV Miscellaneous discussion between conventional and dual symmetric point
of view

To compare conventional and dual symmetric theories of axion electrodynamics,
we derive the second order D’Alembert equation governing the axion induced
electromagnetic field $\mathbf{E}_{ind}$, $\mathbf{B}_{ind}$. In the
literature, the second partial differential equations for induced fields are
more practically used than general solution; therefore, it is reasonable to
write them down in dual symmetric version of axion electrodynamics equations.
First, it is obvious that the applied and radiation fields must be solutions
of

$\displaystyle\square\mathbf{E}_{0}$ $\displaystyle=$
$\displaystyle-\dfrac{\pmb{\nabla}\rho_{e}}{\varepsilon_{0}}-\mu_{0}\dfrac{\partial\mathbf{J}_{e}}{\partial
t},$ (56a) $\displaystyle\square\mathbf{B}_{0}$ $\displaystyle=$
$\displaystyle\mu_{0}\pmb{\nabla}\times\mathbf{J}_{e},$ (56b)

Now, we aim to derive the second order D’Alembert equation governing the axion
induced electromagnetic field $\mathbf{E}_{ind}$, $\mathbf{B}_{ind}$ in
haloscope experiment for both dual symmetric and conventional scenarios.

###  IV.1 In dual symmetric scenario of haloscope experiment

Taking D’Alembert operator $\square$ act on first-order fields
$\mathbf{E}_{ind}$ and $\mathbf{B}_{ind}$ given in Equation (18) leads to the
following equations

$\displaystyle\square\left[\mathbf{E}_{ind}^{dual}-cg_{a\gamma\gamma}a\mathbf{B}_{0}\right]$
$\displaystyle\approx\square\mathbf{E}_{rad}=0,$ (57a)
$\displaystyle\square\left[\mathbf{B}_{ind}^{dual}+\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}\right]$
$\displaystyle\approx\square\mathbf{B}_{rad}=0.$ (57b)

Equation (7) and Equation (56) are used and second order terms of coupling
constant $g_{a\gamma\gamma}$ are neglected. In the static $DC$ haloscope
experiments, applied fields $\mathbf{E}_{0}$ and $\mathbf{B}_{0}$ are static
i.e time-independent. They are induced from surface electric current
$\mathbf{J}_{e}$ and surface electric charge $\rho_{e}$ on cavity ’s surface.
Therefore, on the free space whereas $\rho_{e}=0$ and
$\mathbf{J}_{e}=\mathbf{0}$, Equations (7), (56), (57) in first order
approximation of dual symmetric scenario becomes

$\displaystyle\left(\square+\dfrac{\omega_{a}^{2}}{c^{2}}\right)a\approx 0,$
(58) $\displaystyle\pmb{\nabla}^{2}\mathbf{E}_{0}=0,$ (59a)
$\displaystyle\pmb{\nabla}^{2}\mathbf{B}_{0}=0,$ (59b)
$\displaystyle\square\left[\mathbf{E}_{ind}^{dual}-cg_{a\gamma\gamma}a\mathbf{B}_{0}\right]\approx
0,$ (59c)
$\displaystyle\square\left[\mathbf{B}_{ind}^{dual}+\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}\right]\approx
0.$ (59d)

###  IV.2 In conventional scenario of haloscope experiment

As we have discussed in Section I, in our point of view, the conventional
scenario of axion electrodynamics is equivalent to the dual symmetric axion
electrodynamics with magnetic monopole

$\displaystyle\rho_{m}^{conv}$
$\displaystyle=cg_{a\gamma\gamma}a\rho_{e}+\dfrac{g_{a\gamma\gamma}}{\mu_{0}c}\left(\mathbf{E}+cg_{a\gamma\gamma}a\mathbf{B}\right)\cdot\pmb{\nabla}a$
(60a) $\displaystyle\mathbf{J}_{m}^{conv}$
$\displaystyle=cg_{a\gamma\gamma}a\mathbf{J}_{e}-\dfrac{g_{a\gamma\gamma}}{c\mu_{0}}\dfrac{\partial
a}{\partial t}\left(\mathbf{E}+cg_{a\gamma\gamma}a\mathbf{B}\right)$ (60b)
$\displaystyle\quad\quad\quad\quad\quad-\dfrac{g_{a\gamma\gamma}}{\mu_{0}}\left(c\mathbf{B}-g_{a\gamma\gamma}a\mathbf{E}\right)\times\pmb{\nabla}a.$

On the free space i.e out of the cavity ’s surface on haloscope experiment,
there are no electric charge $\rho_{e}=0$ or current
$\mathbf{J}_{e}=\mathbf{0}$, then the conventional magnetic monopole sources
in the first order approximation are given as follows:

$\displaystyle\rho_{m}^{conv}$
$\displaystyle\approx\dfrac{g_{a\gamma\gamma}}{\mu_{0}c}\mathbf{E}_{0}\cdot\pmb{\nabla}a$
(61a) $\displaystyle\mathbf{J}_{m}^{conv}$
$\displaystyle\approx-\dfrac{g_{a\gamma\gamma}c}{\mu_{0}}\left[\dfrac{\partial
a}{\partial
t}\dfrac{\mathbf{E}_{0}}{c^{2}}+\mathbf{B}_{0}\times\pmb{\nabla}a\right].$
(61b)

To obtain the D’Alembert equations for axion-induced electromagnetic field in
this conventional scenario $\mathbf{E}_{ind}^{conv}$ and
$\mathbf{B}_{ind}^{conv}$, we act the curl operator $\pmb{\nabla}\times$ on
Equations (3c) and (3d) followed by using the identity
$\pmb{\nabla}\times\pmb{\nabla}\times\mathbf{A}=\pmb{\nabla}\left(\pmb{\nabla}\cdot\mathbf{A}\right)-\pmb{\nabla}^{2}\mathbf{A}$
as well as Equations (3a) and (3b). Finally the D’Alembert equations for
axion-induced electromagnetic field in this conventional scenario are

$\displaystyle\square\left[\mathbf{E}_{ind}^{conv}-cg_{a\gamma\gamma}a\mathbf{B}_{0}\right]$
$\displaystyle\approx-\mu_{0}\pmb{\nabla}\times\mathbf{J}_{m}^{conv},$ (62a)
$\displaystyle\square\left[\mathbf{B}_{ind}^{conv}+\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}\right]$
$\displaystyle\approx-\mu_{0}\left[\pmb{\nabla}\rho_{m}^{conv}+\dfrac{\partial\mathbf{J}_{m}^{conv}}{c^{2}\partial
t}\right].$ (62b)

Equation (62) actually comes from the generalization of Equation (12) with
non-zero magnetic monopole.

For free axion, the propagation of axion field is the plane wave
$a(\mathbf{r},t)=a_{0}(\mathbf{k},\omega)\exp\left[\imath\left(\mathbf{k}\cdot\mathbf{r}-\omega
t\right)\right]$. If we write the wave vector $\mathbf{k}$ as follow
$\mathbf{k}=\dfrac{\omega}{c}\vec{n}$ then the module of $\vec{n}$ is
$|\vec{n}|=v_{a}/c$. $\vec{n}$ is the dimensionless quantity whose module
relates to the relative speed of free axion particles. For cold axion or
cosmology axion, there velocity is often small $v_{a}/c\sim 10^{-3}$ [42, 43,
44, 45]. Then the resonant mainly occurs near the ground state of axion
$\omega\approx\omega_{a}$. The gradient and time derivative of axion field are
approximately

$\pmb{\nabla}a\approx\imath\omega_{a}ac^{-1}\mathbf{n},\quad\partial
a/\partial t=-\imath\omega_{a}a.$ (63)

Consequently the conventional magnetic monopole sources in the first order
approximation becomes:

$\displaystyle\rho_{m}^{conv}$
$\displaystyle\approx\dfrac{\imath\omega_{a}g_{a\gamma\gamma}a}{\mu_{0}c^{2}}\mathbf{E}_{0}\cdot\mathbf{n}$
(64a) $\displaystyle\mathbf{J}_{m}^{conv}$
$\displaystyle\approx\dfrac{\imath\omega_{a}g_{a\gamma\gamma}a}{\mu_{0}}\left[\mathbf{n}\times\mathbf{B}_{0}+\dfrac{\mathbf{E}_{0}}{c}\right].$
(64b)

Substituting Equations (64) into Equations (62), keeping the first order of
axion field or axion velocity and noticing that $\square
a=-\omega_{a}^{2}c^{-2}a$, we finally obtain the D’Alembert equations governs
free propagation of axion-induced electromagnetic fields in the conventional
scenario of axion haloscope experiments

$\displaystyle\square\left[\mathbf{E}_{ind}^{conv}-cg_{a\gamma\gamma}a\mathbf{B}_{0}+g_{a\gamma\gamma}a\mathbf{n}\times\mathbf{E}_{0}\right]$
$\displaystyle\approx 0,$ (65a)
$\displaystyle\square\left[\mathbf{B}_{ind}^{conv}-g_{a\gamma\gamma}a\mathbf{n}\times\mathbf{B}_{0}\right]$
$\displaystyle\approx 0.$ (65b)

###  IV.3 Comparision between conventional and dual symmetric point of view in
axion haloscope experiments

As can be seen from Equations (57) and (65), the axion-induced electromagnetic
in haloscope experiments in both theories are sightly different. While the
axion-induced fields in dual symmetric scenario are simply

$\displaystyle\mathbf{E}_{ind}^{dual}\approx
cg_{a\gamma\gamma}a\mathbf{B}_{0}+\mathbf{E}_{rad}^{dual},$ (66a)
$\displaystyle\mathbf{B}_{ind}^{dual}\approx-\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}+\mathbf{B}_{rad}^{dual},$
(66b)

the one in conventional scenario are more complicated

$\displaystyle\mathbf{E}_{ind}^{conv}\approx
cg_{a\gamma\gamma}a\mathbf{B}_{0}-g_{a\gamma\gamma}a\mathbf{n}\times\mathbf{E}_{0}+\mathbf{E}_{rad}^{conv},$
(67a) $\displaystyle\mathbf{B}_{ind}^{conv}\approx
g_{a\gamma\gamma}a\mathbf{n}\times\mathbf{B}_{0}+\mathbf{B}_{rad}^{conv}.$
(67b)

where $\mathbf{E}_{rad},\mathbf{B}_{rad}$ is electromagnetic radiation arising
from boundary effect of haloscope cavity. For simplicity, in further
investigation, we only consider PEC cavity in which the induced electric field
vanishes at the surface $C$ of the cavity: $\mathbf{E}_{ind}(\mathbf{r}\in
C)=0$. Then the differences between dual symmetric and conventional theories
are

$\displaystyle\delta\mathbf{E}_{ind}$
$\displaystyle=\mathbf{E}_{ind}^{conv}-\mathbf{E}_{ind}^{dual}$ (68a)
$\displaystyle\approx-
g_{a\gamma\gamma}a\mathbf{n}\times\mathbf{E}_{0}+\delta\mathbf{E}_{rad},$
$\displaystyle\delta\mathbf{B}_{ind}$
$\displaystyle=\mathbf{B}_{ind}^{conv}-\mathbf{B}_{ind}^{dual}$ (68b)
$\displaystyle\approx\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}+g_{a\gamma\gamma}a\mathbf{n}\times\mathbf{B}_{0}+\delta\mathbf{B}_{rad}.$

From boundary condition of PEC cavity, we can estimate the order of different
radiation terms
$\delta\mathbf{E}_{rad}=\mathbf{E}_{rad}^{conv}-\mathbf{E}_{rad}^{dual}$ and
$\delta\mathbf{B}_{rad}=\mathbf{B}_{rad}^{conv}-\mathbf{B}_{rad}^{dual}$ are
$\delta E_{rad}\sim g_{a\gamma\gamma}a_{0}v_{a}c^{-1}E_{0}$ and $\delta
B_{rad}\sim c^{-1}\delta E_{rad}$ respectively. Now we start to examine the
difference between two theories some specific cases of axion haloscope
experiments.

First of all, we examine the traditional axion haloscope experiments using
applied DC magnetic field $\mathbf{B}_{0}\neq 0$ on a resonant cavity without
any electric field $\mathbf{E}_{0}=0$. Then $\delta E_{rad},\delta B_{rad}\to
0$ and hence, the differences between dual symmetric and conventional theories
only appears in the axion-induced magnetic field and its magnitude is
proportional to axion velocity $v_{a}$:

$\displaystyle\delta\mathbf{E}_{ind}=0,$ (69a)
$\displaystyle\delta\mathbf{B}_{ind}=g_{a\gamma\gamma}a\mathbf{n}\times\mathbf{B}_{0}\sim
g_{a\gamma\gamma}av_{a}c^{-1}B_{0}.$ (69b)

Under the LWA region, $\pmb{\nabla}a\to 0$ or $v_{a}\to 0$, both theories
coincides to each other: $\delta\mathbf{E}_{ind}=0$,
$\delta\mathbf{B}_{ind}=0$. This is confirmed since the solution of
conventional axion Maxwell equations under LWA approxiation in previous works
[18, 23] is the same as ours in Equations (24). The equivalence between two
theories in LWA region is understandable since the required monopole sources
in free space for conventional theory are proportional to gradient of axion
field
$\rho_{m}^{conv}=\mu_{0}^{-1}g_{a\gamma\gamma}\mathbf{E}_{0}\cdot\pmb{\nabla}a\to
0$,
$\mathbf{J}_{m}^{conv}=\mu_{0}^{-1}g_{a\gamma\gamma}c\mathbf{B}_{0}\times\pmb{\nabla}a\to
0$ and hence, become zero in LWA region $\pmb{\nabla}a\to 0$. Another case in
which the difference between two theories vanishes is when the axion source is
homogeneous and the cavity is symmetry. In this case, the contribution of
axion particles in one direction $\mathbf{n}$ will be cancelled by the
contribution of axion particles in opposite direction $-\mathbf{n}$. This
could happen when using resonant cavity such as cylindrical solenoid,
spherical solenoid, two-parallel-sheet cavity, toroidal solenoid with a
rectangular cross-section, or with a circular cross-section.

Next, from above observations, we consider two other possibilities of
detecting the differences between dual symmetric and conventional scenarios of
axion electrodynamics. The first possible experiments is using directional
axion detection [42, 43, 44, 45]. Enhancement of the directional sensitivity
by using asymmetric cavity such as long tube or spheroidal cavities, we can
keep the different axion-induced magnetic field non zero
$\delta\mathbf{B}_{ind}\sim g_{a\gamma\gamma}av_{a}c^{-1}B_{0}$. Another
possible set up is applying an DC electric field $\mathbf{E}_{0}\neq 0$ on
axion haloscope cavity. To the best of our knowledge, there are no proposals
about applying external DC electric field in haloscope experiment to distinct
between dual symmetric and conventional scenarios of axion electrodynamics.
Even when searching axion particles in LWA region or using symmetric cavity,
axion-induce fields calculated from two theories are still sightly different

$\displaystyle\delta\mathbf{E}_{ind}\approx 0,$ (70a)
$\displaystyle\delta\mathbf{B}_{ind}\approx\dfrac{g_{a\gamma\gamma}a}{c}\mathbf{E}_{0}\sim
g_{a\gamma\gamma}a_{0}c^{-1}E_{0}.$ (70b)

If we using high enough DC electric field $E_{0}>v_{a}B_{0}>10^{-3}cB_{0}$,
the distinguish between two theories is much more sensitive than directional
axion detection.

Before ending our discussion, we have to emphasize again that the compact
solution of axion-induced fields in dual symmetric theory (57) are not only
correct for static axion haloscope experiments but also for the AC ones. Thus,
dual symmetric theory of axion electrodynamics provided potential benchmark
for various types of axion haloscope experiments.

##  V Conclusion

In this paper, we have considered the axion electrodynamics in haloscope
experiments under the perspective of dual symmetry. First, the dual symmetric
revision of the axion Maxwell equation allows us to obtain an exact analytical
solution in haloscopes without LWA approximation. Moreover, we pointed out
that our solution from the dual symmetric theory is identical to the one from
conventional theory when the wavelength of the axion is in the long-wavelength
regime. It can be explained by the fact that required monopole sources in free
space for conventional theory are proportional to gradient of axion field i.e
from Equation (5):
$\rho_{m}^{conv}=\mu_{0}^{-1}g_{a\gamma\gamma}\mathbf{E}_{0}\cdot\pmb{\nabla}a\to
0$,
$\mathbf{J}_{m}^{conv}=\mu_{0}^{-1}g_{a\gamma\gamma}c\mathbf{B}_{0}\times\pmb{\nabla}a\to
0$, then the magnetic monopole sources are such small that can be ignored.
Hence, the dual symmetry approach examined in our work provides the same
results as the conventional approach in Reference [11] without using LWA
approximation. Particularly, we apply to (im)perfect electric conductive
resonant cavity in various shapes such as cylinder, two parallel sheets,
sphere, circular torus, and rectangular torus. The characteristic quantities
of these geometries like form factors related to summation/difference of
induced electric and magnetic stored energy and their ratio have also been
examined. The resonance of the axion-to-photon conversion power
$P_{a\to\gamma\gamma}$ in imperfect electric conductive and the zeros of the
ratio $\mathfrak{R}$ are also observed. This research may support scientists
and technicians in the field regarding the search for axions or axion-like
particles and provide more precise analytical frame to characterize axion
dynamics in haloscope experiments with different geometries of resonant
cavities. Finally, we also discussed about the difference bewteen dual
symmetric and conventional theories of axion haloscope experiments. Our
discussion could suggest some possibilities of distinguish these two theories
by using known directional axion detection or external DC electric field on
haloscopes.

## Acknowledgement

One of the author, Dai-Nam Le, thanks to Dr. Eibun Senaha (Science and
Technology Advanced Institute, Van Lang University, Vietnam) for his helpful
discussion when finalizing this work. Dai-Nam Le was funded by Vingroup Joint
Stock Company and supported by the Domestic Master/ PhD Scholarship Programme
of Vingroup Innovation Foundation (VINIF), Vingroup Big Data Institute
(VINBIGDATA), code VINIF.2020.TS.03. The authors also thank the referees for
their constructive comments.

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##  Appendix A The spatial distribution of axion-induced electromagnetic
stored energy density

####  A.0.1 Infinite cylindrical solenoid

Figure 14: (a,b,c) The spatial distribution of axion-induced electromagnetic
stored energy density inside and outside the cavity when
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$.

####  A.0.2 Spherical solenoid

Figure 15: (a,b,c) The spatial distribution of axion-induced electromagnetic
stored energy density inside and outside the cavity when
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$ at equator of
spherical cavity $\theta=0$.

####  A.0.3 Infinite two-parallel-sheet cavity

Figure 16: (a,b,c) The spatial distribution of axion-induced electromagnetic
stored energy density inside and outside the cavity when
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$.

####  A.0.4 Toroidal solenoid with rectangular cross section

Figure 17: (a,b,c) The spatial distribution of axion-induced electromagnetic
stored energy density inside and outside the cavity when
$\omega_{a}=1.1\omega_{c,1},1.1\omega_{c,2},1.1\omega_{c,3}$.

####  A.0.5 Toroidal solenoid with circular cross section

Figure 18: (a,b,c) The spatial distribution of axion-induced electromagnetic
stored energy density inside and outside the cavity when
$\omega_{a}=1.2\omega_{c,1},1.2\omega_{c,2},1.2\omega_{c,3}$.