# Axionic and nonaxionic electrodynamics in plane and circular geometry

Iver Brevik1 1Department of Energy and Process Engineering, Norwegian
University of Science and Technology, N-7491 Trondheim, Norway  Amedeo M.
Favitta2 2 Department of Physics, University of Palermo, I-90123 Palermo,
Italy  Masud Chaichian3 3Department of Physics, University of Helsinki, O. O.
Box 64, FI-00014 Helsinki, Finland

###### Abstract

Various aspects of axion electrodynamics in the presence of a homogeneous and
isotropic dielectric medium are discussed. 1. We consider first the ”antenna-
like” property of a planar dielectric surface in axion electrodynamics,
elaborating on the treatment given earlier on this topic by Millar et al.
(2017). We calculate the electromagnetic energy transmission coefficient for a
dielectric plate, and compare with the conventional expression in ordinary
electrodynamics. 2. We consider the situation where the medium exterior to the
plate, assumed elastic, is ”bent back” and glued together, so that we obtain a
circular dielectric string in which the waves can propagate clockwise or
counterclockwise. As will be shown, a stationary wave pattern is permitted by
the formalism, and we show how the amplitudes for the two counterpropagating
waves can be found. 3. As a special case, by omitting axions for a moment, we
analyze the Casimir effect for the string, showing its similarity as well as
its difference with the Casimir effect of a scalar field for a piecewise
uniform string (Brevik and Nielsen 1990). 4. Finally, including axions again
we analyze the enhancement of the surface-generated electromagnetic radiation
near the center of a cylindrical haloscope, where the interior region is a
vacuum and the exterior region a metal. This enhancement is caused by the
curvature of the boundary, and is mathematically a consequence of the behavior
of the Hankel function of the second kind for small arguments. A simple
estimate shows that enhancement may be quite significant, and can therefore be
of experimental interest. This proposal is suggested as an alternative to the
reflector arrangement in a similar arrangement recently discussed by Liu et
al. (2022).

##  I Introduction

The ”antenna-like” behavior of a single dielectric plane surface in axion
electrodynamics in the presence of a strong external magnetic field is a
rather remarkable phenomenon. In particular, it is quite perplexing that the
Poynting vector normal to a dielectric surface is no longer continuous across
the surface; a property otherwise considered as a cornerstone in
electromagnetic theory. There occurs an extra electromagnetic radiation of
energy from the surface. Of course, this energy cannot stem from ”nowhere”. It
is rather a consequence of the interchange of field energies interchanged
between two reservoirs, namely the usual electromagnetic reservoir, and the
other one due to tha axion field. A detailed exposition of these properties of
axion electrodynamics is given by Millar et al. [1]. The effect is also
planned use of in experimental broadband tests [2].

Our purpose with the present paper is first to elaborate further on some of
the consequences of this dielectric-surface effect. After giving a brief
survey of axion electrodynamic formalism in the next section, we consider in
Sec. III the most simple dielectric system where two surfaces are involved,
namely the planar dielectric disk. We point out the close similarity between
the transmission coefficients in the axion case and in the non-axion case, a
similarity that is not quite trivial. Thereafter in Sec. IV, still dealing
with two-surface systems, we turn the exterior medium 2 ”back” and join it
with medium 1 so that we obtain a circular loop or string, and investigate if
the axion formalism admits a stationary state implying clockwise and
counterclockwise propagating modes. Actually the formalism does so. Such a
system is most likely only of fundamental interest, but the positive outcome
of the analysis demonstrates the flexibility of the axionic formalism. As a
byproduct, if we for a moment ignore the axion field we can calculate the
electromagnetic Casimir energy of the system. This is considered in Sec. V. We
show how the systems bears similarities to the piecewise uniform relativistic
string which has been considered repeatedly earlier [3, 4, 5, 6, 7, 8, 9].
Finally, in Sec. VI we return to the antenna-like property of dielectric
surfaces, and investigate the enhancement of the electric field near the
center of a cylindrical haloscope caused by the curvature of the emitting
boundary. Here the central region is a vacuum, while the outer region is a
metal. The enhancement can actually be quite significant. This opens for the
possible measurement of the emitted field from the cylindrical boundary, and
may thus be an alternative to the broadband solenoidal haloscope recently
proposed by Liu et al. [2].

Some of the pioneering papers on axion electrodynamics are listed in Refs.
[10, 11, 13, 12, 14, 15, 16, 17, 18]. More recent works can be found in Refs.
[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43].

##  II Basic formalism in a dielectric environment

We consider a pseudoscalar axion $a=a({\bf r},t)$ present in the entire
universe, making a two-photon interaction with the electromagnetic field. We
assume a dielectric environment where the permittivity is $\varepsilon$ and
the permeability is $\mu$, where these material parameters are constants. The
constitutive relations are $\bf{D}=\varepsilon{\bf E},{\bf B}=\mu\bf H$. There
are two field tensors, $F_{\alpha\beta}$ and $H_{\alpha\beta}$, where $\alpha$
and $\beta$ run from 0 to 3. We assume the standard Minkowski space with the
convention $g_{00}=-1$. The dual is defined as
$\tilde{F}^{\alpha\beta}=\frac{1}{2}\varepsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}$,
with $\varepsilon^{0123}=1$.

For convenience we give the expression for the field tensors explicitly,

$F_{\alpha\beta}=\left(\begin{array}[]{rrrr}0&-E_{x}&-E_{y}&-E_{z}\\\
E_{x}&~{}0&B_{z}&-B_{y}\\\ E_{y}&-B_{z}&~{}0&B_{x}\\\
E_{z}&B_{y}&-B_{x}&~{}0\end{array}\right),$ (1)
$H^{\alpha\beta}=\left(\begin{array}[]{rrrr}0&D_{x}&D_{y}&D_{z}\\\
-D_{x}&0&H_{z}&-H_{y}\\\ -D_{y}&-H_{z}&0&H_{x}\\\
-D_{z}&H_{y}&-H_{x}&0\end{array}\right).$ (2)

The Lagrangian is

${\cal{L}}=-\frac{1}{4}F_{\alpha\beta}{H}^{\alpha\beta}+{\bf A\cdot
J}-\rho\Phi-\frac{1}{2}\partial_{\mu}a\partial^{\mu}a-\frac{1}{2}m_{a}^{2}a^{2}-\frac{1}{4}g_{\gamma}\frac{\alpha}{\pi}\frac{1}{f_{a}}a(x)F_{\alpha\beta}\tilde{F}^{\alpha\beta}.$
(3)

Here, $\rho$ and $\bf J$ are the usual electromagnetic charge and current
densities (since the axions are electrically neutral they can not contribute);
$g_{\gamma}$ is a model-dependent constant for which we adopt the value $0.36$
[22]; $\alpha$ is the fine structure constant, and $f_{a}$ is the axion decay
constant whose value is insufficiently known. Often it is assumed that
$f_{a}\sim 10^{12}~{}$GeV.

Defining the combined axion-two-photon coupling constant as

$g_{a\gamma\gamma}=g_{\gamma}\frac{\alpha}{\pi}\frac{1}{f_{a}},$ (4)

we see that the last term in the Lagrangian (3) can be written as
${\cal{L}}_{a\gamma\gamma}=g_{a\gamma\gamma}a(x)\,{\bf E\cdot B}.$

It is convenient to define the quantity $\theta(x)$,

$\theta(x)=g_{a\gamma\gamma}a(x).$ (5)

Based on the expression (3), the extended Maxwell equations can then be
written as

${\bf\nabla\cdot D}=\rho-{\bf B\cdot\nabla}\theta,$ (6) ${\bf\nabla\times
H}={\bf J}+\dot{\bf D}+\dot{\theta}{\bf B}+{\bf\nabla}\theta\times{\bf E},$
(7) ${\bf\nabla\cdot B}=0,$ (8) ${\bf\nabla\times E}=-\dot{\bf B}.$ (9)

These equations are general, i.e., there are no restrictions so far on the
spacetime variation of $a(x)$. The equations are moreover relativistic
covariant, with respect to shift of the inertial system.

The governing equations for the fields can correspondingly be written as

$\nabla^{2}{\bf E}-\varepsilon\mu\ddot{\bf E}={\bf\nabla(\nabla\cdot
E)}+\mu\dot{\bf J}+\mu\frac{\partial}{\partial t}\left[\dot{\theta}{\bf
B}+{\bf\nabla}\theta{\bf\times E}\right],$ (10) $\nabla^{2}{\bf
H}-\varepsilon\mu\ddot{\bf H}=-{\bf\nabla\times
J}-{\bf\nabla\times}[\dot{\theta}{\bf B}+{\bf\nabla}\theta{\bf\times E}].$
(11)

The dynamical field entities are here the electromagnetic fields; we limit
ourselves to a perturbative approach in which the axions are the perturbation.
We do not consider the field equations for the axions explicitly.

Assume now that a strong static magnetic field ${\bf B}_{e}=B_{e}\hat{\bf z}$
acts in a region where $\rho$ and $\bf J$ are zero and the axion field is
spatially uniform but varies harmonically in time,

$a(t)=a_{0}\cos\omega_{a}t.$ (12)

This is the situation usually found in the inner region of a haloscope. Then
it is convenient to separate out the part of $\bf E$ that is caused by the
uniformly fluctuating axions. Calling this contribution ${\bf E}_{a}(t)$, we
see from Eq. (10) that it is connected by the $\ddot{\theta}$ term. From the
governing equation for ${\bf E}_{a}(t)$,

$\nabla^{2}{\bf E}_{a}-\varepsilon\mu\ddot{\bf E}_{a}=\mu\ddot{\theta}{\bf
B}_{e}$ (13)

we then obtain, after omitting the $\nabla^{2}$ term, the solution

${\bf E}_{a}(t)=-\frac{1}{\varepsilon}E_{0}\cos\omega_{a}t\,\hat{\bf z},$ (14)

where

$E_{0}=\theta_{0}B_{e}.$ (15)

After the separation of the component ${\bf E}_{a}$, the field equation (10)
takes the reduced form

$\nabla^{2}{\bf E}-\varepsilon\mu\ddot{\bf E}={\bf\nabla(\nabla\cdot
E)}+\mu{\dot{\bf J}}+\mu[\dot{\theta}\dot{\bf
B}+{\bf\nabla}\theta{\bf\times\dot{E}}].$ (16)

Likewise for the magnetic field

$\nabla^{2}{\bf H}-\varepsilon\mu\ddot{\bf H}=-{\bf\nabla\times
J}-\left[\dot{\theta}{\bf\nabla\times B}+({\bf\nabla}\theta){\bf\nabla\cdot
E}-({\bf\nabla}\theta\cdot{\bf\nabla}){\bf E}\right].$ (17)

##  III Antenna-like behavior: two plane parallel surfaces

We begin by considering one single planar dielectric surface, placed at $x=0$,
separating the left region 1 (refractive index $n_{1}$) from the right region
2 (refractive index $n_{2}$). We assume the simple case where the media are
nonmagnetic and $n_{1}$, $n_{2}$ constants and real. A strong static magnetic
field ${\bf B}_{e}=B_{e}\hat{\bf z}$ is imposed in the $z$ direction. An
incoming wave polarized in the $z$ direction comes in from the left,
propagates in the $x$ direction, and becomes partly reflected by the surface.
The basic property following from the extended Maxwell equations in the
present context is that the components of $\bf E$ and $\bf H$ parallel to the
surface have to be continuous (as in ordinary electrodynamics) at $x=0$.

This is the situation analyzed in detail by Millar et al. [1]. Because of the
axions there will be produced two outgoing electromagnetic waves, one going to
the left and one going to the right. In this sense we can consider the
dielectric surface to have ”antenna-like” properties. It does not mean that
electromagnetic energy comes out from nothing, however, but that there occurs
an energy interchange between the ordinary electromagnetic and the axion
reservoirs.

It is now convenient, following Ref. [1], to distinguish the produced
traveling fields by an extra index $\gamma$. Continuity of ${\bf
E}_{\parallel}$ gives straightaway

$E_{1}^{\gamma}+E_{1}^{a}=E_{2}^{\gamma}+E_{2}^{a}.$ (18)

Next, we may take into account the relationship

$H^{\gamma}=\pm\frac{1}{n}E^{\gamma},$ (19)

which holds for a propagating wave when the medium is nonmagnetic. This
implies that the boundary condition for ${\bf H}_{\parallel}$ can be written
as

$-n_{1}E_{1}^{\gamma}=n_{2}E_{2}^{\gamma},$ (20)

expressing that the two wave vectors ${\bf k}_{1}$ and ${\bf k}_{2}$ are
antiparallel. As $E^{a}=-(1/\varepsilon)E_{0}$, we can then solve for the
produced fields to get

$E_{1}^{\gamma}=-\frac{E_{0}}{n_{1}}\left(\frac{1}{n_{2}}-\frac{1}{n_{1}}\right),\quad
E_{2}^{\gamma}=\frac{E_{0}}{n_{2}}\left(\frac{1}{n_{2}}-\frac{1}{n_{1}}\right),$
(21)

(recall that $E_{0}=\theta_{0}B_{e}$).

Remark: The argument above did not imply there to be any infalling initial
photons from the left. This is the most fundamental constellation. The
argument however implies a nontrivial point, namely the use of Eq. (19) in an
initial vacuum. This equation is not a general one in electrodynamics, but is
implicitly resting on the assumption that there are propagating waves. This
point thus brings some concern about the validity of the argument in this
special case where there initially a vacuum. Now, Millar et al. discuss also
the more general case where there are infalling photons from the left, and in
that case the use of Eq. (19) is of course nonproblematic.

Now move on to consider the situation with two dielectric surfaces; we focus
first on simple dielectric slab of thickness $d$,surrounded by vacuum. The
refractive index is $n=\sqrt{\varepsilon}$, similarly as above. It is of
interest to consider the energy transmission coefficient

$T=\big{|}\frac{E_{T}}{E_{I}}\big{|}^{2}$ (22)

in the presence of axions, $E_{T}$ and $E_{I}$ referring to the transmitted
and incident wave amplitudes. Based upon Eq. (5.4a) in Ref. [1] we obtain

$T=\frac{4n^{2}}{4n^{2}+(n^{2}-1)^{2}\sin^{2}kd},$ (23)

where $k=2\pi/\lambda=n\omega$. Remarkably enough, this expression does not
contain $E_{0}$. Some special cases are noteworthy: if $d=\lambda/4$ the
transmission is at minimum, $T=T_{\rm min}=4n^{2}/(n^{2}+1)^{2}$, whereas if
$d=\lambda/2$, the transmission is at maximum, $T=T_{\rm max}=1$.

It is of interest to compare this transmission coefficient from that occurring
in ordinary electrodynamics. Giving the latter quantity an extra subscript
elmag, we obtain (page 514 in Stratton [44])

$T_{\rm elmag}=\frac{4n^{2}}{(n^{2}+1)^{2}-(n^{2}-1)^{2}\sin^{2}kd}.$ (24)

It is seen that $T$ and $T_{\rm elmag}$ are different, what could be expected
since their derivations are different. Now, if $d=\lambda/4$, $T_{\rm
elmag}=1$, while if $d=\lambda/2$, $T_{\rm elmag}=4n^{2}/(n^{2}+1)^{2}$, its
minimum value.

##  IV The closed string geometry

In this section we will continue to consider dielectric systems containing two
interfaces separating media of refractive indices $n_{1}$ and $n_{2}$, but now
in a form that has not been considered before as far as we know. Let $n_{1}$
refer to the left medium, $n_{2}$ refer to the second medium, and assume that
the media are elastic so that medium 2 can be turned back and glued to the
left side of medium 1. Therewith we obtain a ring-formed system. One may ask:
does such a system allow stationary oscillations to occur when axions are
present? It is of physical interest to examine this point, not because of the
applicability of the formalism in practice, but rather as a test of the
flexibility of this kind of generalized electrodynamics.

Figure 1: Geometry and notation of the closed string.

Figure 1 shows the configuration. We let $\sigma$ denote the length coordinate
along the string, such that the two dielectric junctions are at $\sigma=0$ and
$\sigma=L_{I}$. The total length of the string is $L=L_{I}+L_{II}$, so that
the junctions $\sigma=0$ and $\sigma=L$ are overlapping. We will be interested
in the fields in the interior regions of the string. The string is lying in
the $xy$ plane, and a strong uniform magnetic field ${\bf B}_{e}$ is applied
in the $z$ direction.

We will seek for stationary oscillations of the electromagnetic oscillations
in the string. If $E_{I}(\sigma,t)$ and $E_{II}(\sigma,t)$ are the electric
fields in the two regions, we have in complex representation

$E_{I}(\sigma,t)=\xi_{I}e^{in_{1}\omega\sigma-i\omega
t}+\eta_{I}e^{-in_{1}\omega\sigma-i\omega t},$ (25)
$E_{II}(\sigma,t)=\xi_{II}e^{in_{2}\omega(\sigma-L_{I})-i\omega
t}+\eta_{II}e^{-in_{2}\omega(\sigma-L_{I})-i\omega t},$ (26)

, where $\xi_{I},\eta_{I},\xi_{II},\eta_{II}$ are constants. Analogously,
using the same relationship $H=\pm nE$ as previously, we have for the magnetic
field

$H_{I}(\sigma,t)=n_{1}\left[\xi_{I}e^{in_{1}\omega\sigma-i\omega
t}-\eta_{I}e^{-in_{1}\omega\sigma-i\omega t}\right],$ (27)
$H_{II}(\sigma,t)=n_{2}\left[\xi_{II}e^{in_{2}(\sigma-L_{I})-i\omega
t}-\eta_{II}e^{-in_{2}\omega(\sigma-L_{I})-i\omega t}\right].$ (28)

We omit the time factor $e^{-i\omega t}$ and introduce the shorthand notation

$\delta_{1}=n_{1}\omega L_{I},\quad\delta_{2}=n_{2}\omega L_{II}.$ (29)

The boundary conditions at the junctions are, for the electric field,

$-\frac{E_{0}}{\varepsilon_{1}}+\xi_{I}e^{i\delta_{1}}+\eta_{I}e^{-i\delta_{1}}=-\frac{E_{0}}{\varepsilon_{2}}+\xi_{II}+\eta_{II},\quad\sigma=L_{I},$
(30)
$-\frac{E_{0}}{\varepsilon_{2}}+\xi_{II}e^{i\delta_{2}}+\eta_{II}e^{-i\delta_{2}}=-\frac{E_{0}}{\varepsilon_{1}}+\xi_{I}+\eta_{I},\quad\sigma=L,$
(31)

and for the magnetic field,

$n_{1}(\xi_{1}e^{i\delta_{1}}-\eta_{I}e^{-i\delta_{1}})=n_{2}(\xi_{II}-\eta_{II}),\quad\sigma=L_{I},$
(32)
$n_{1}(\xi_{I}-\eta_{I})=n_{2}(\xi_{II}e^{i\delta_{2}}-\eta_{II}e^{-i\delta_{2}}),\quad\sigma=L,$
(33)

where $E_{0}=\theta_{0}B_{e}=g_{a\gamma\gamma}a_{0}B_{e}$ as before.

We introduce the symbol $x$ for the refractive index ratio,

$x=\frac{n_{1}}{n_{2}},$ (34)

and consider the scheme

$\left(\begin{array}[]{rrrr}e^{i\delta_{1}}&e^{-i\delta_{1}}&-1&-1\\\
1&1&-e^{i\delta_{2}}&-e^{-i\delta_{2}}\\\
xe^{i\delta_{1}}&-xe^{-i\delta_{1}}&-1&1\\\
x&-x&-e^{i\delta_{2}}&e^{-i\delta_{2}}\end{array}\right)\left(\begin{array}[]{c}\xi_{I}\\\
\eta_{I}\\\ \xi_{II}\\\
\eta_{II}\end{array}\right)=\left(\begin{array}[]{c}E_{0}\left(\frac{1}{\varepsilon_{1}}-\frac{1}{\varepsilon_{2}}\right)\\\
E_{0}\left(\frac{1}{\varepsilon_{1}}-\frac{1}{\varepsilon_{2}}\right)\\\ 0\\\
0\end{array}\right).$ (35)

Here the determinant $D$ of the system matrix $M_{ik}$ can be calculated to be

$D=\det
M_{ik}=-8x+2(1+x)^{2}\cos(\delta_{1}+\delta_{2})-2(1-x)^{2}\cos(\delta_{1}-\delta_{2}).$
(36)

This is a real quantity. We can now calculate explicit expressions for the
field amplitudes $\xi_{I},\eta_{I},\xi_{II},\eta_{II}$ ,in the two regions of
the string. The expressions become complicated, and will not be given here.
There is one particular case of interest, however, namely when $n_{2}$ becomes
large in comparison to $n_{1}$ so that the ratio $x\rightarrow 0$. The lengths
$L_{I}$ and $L_{II}$ are assumed arbitrary. From Eq. (36) we obtain in this
case

$D(x\rightarrow
0)=2\cos(\delta_{1}+\delta_{2})-2\cos(\delta_{1}-\delta_{2})=-4\sin\delta_{1}\sin\delta_{2},$
(37)

and after some calculation we obtain, as an example, the first of the
amplitudes in the region $0<\sigma<L_{I}$ (note that
$1/\varepsilon_{2}\rightarrow 0$),

$\xi_{I}=\frac{E_{0}}{2\varepsilon_{1}}\left[1-i\frac{1-\cos\delta_{1}}{\sin\delta_{1}}\right],\quad
x\rightarrow 0.$ (38)

It is noteworthy that this expression does not contain the phase $\delta_{2}$
related to the length $L_{II}$. It is of further interest to consider the case
where the length $L_{I}\rightarrow 0$, corresponding to a kind of point defect
sitting on an otherwise uniform string. As $\delta_{1}\rightarrow 0$ in this
case, we see from the last equation that

$\xi_{I}=\frac{E_{0}}{2\varepsilon_{1}},\quad x\rightarrow 0,L_{I}\rightarrow
0,$ (39)

which is a real quantity. If $E_{0}=0$, the axion-induced forced oscillations
vanish.

In conclusion, we have managed to show that the axionic electrodynamic scheme
is flexible enough to upheld stationary oscillations in the closed string
geometry.

##  V Casimir effect for the closed string

We now put $E_{0}=0$, so that the forced axion-induced oscillations vanish,
and those remaining possible are only the free oscillations. They correspond
to the system determinant $D$ being zero. For a given set of geometric
quantities $L_{I},L_{II},n_{1},n_{2}$, the eigenfrequencies $\omega$ are thus
found by solving $D=0$ using Eq. (36). Moreover, this geometry makes it very
natural to consider the Casimir effect also, after applying an appropriate
regularization. This will be the topic of the present section. Actually, this
situation bears a considerable similarity with the Casimir energy for the
relativistic piecewise uniform string, as mentioned earlier [3, 4, 5, 6, 7, 8,
9].

Before embarking on the general case, it is convenient to start with special
cases of interest. The simplest case is when the sting is uniform, $x=1$,
corresponding to $n_{1}=n_{2}=n$. Then,

$D(x=1)=-8(1-\cos\omega nL),$ (40)

so that the eigenfrequencies become

$\omega_{N}=\frac{2\pi N}{nL},$ (41)

with $N=1,2,3,...$ Assuming implicitly that we apply a regularization
procedure, for instance that of a cutoff regularization, we can express the
Casimir energy as

$E_{\rm uniform}=2\times\frac{1}{2}\sum_{N=1}^{\infty}\omega_{N},$ (42)

where the factor 2 in front accounts for the degeneracy of the left-right
running modes. The result is (cf., for instance, Ref. [3])

$E_{\rm uniform}=-\frac{\pi}{6L}.$ (43)

. The next special case of interest is when $x\rightarrow 0$, the case
considered also in the previous section (note that the value of $n_{1}$ itself
is arbitrary, not necessarily small). Then, from Eq. (37) we see that the
eigenfrequencies occur in two branches,

$\omega_{N}=\frac{\pi N}{n_{1}L_{I}},$ (44) $\omega_{N}=\frac{\pi
N}{n_{2}L_{II}},$ (45)

again with $N=1,2,3,...$ (there is no degeneracy in this case, thus no extra
factor 2). Introducing the symbol $s$ for the length ratio,

$s=\frac{L_{II}}{L_{I}},$ (46)

we get in this case

$E(x\rightarrow 0)=-\frac{\pi}{24L}\left(s+\frac{1}{s}-2\right).$ (47)

This Casimir energy is in general negative, but attains its maximum value zero
when the pieces have equal lengths, $s=1$.

In the general case when $x$ is arbitrary the situation is more complex, but
can yet be handled in a reasonable simple way by making use of the argument
principle. This principle says that any meromorphic function $g(\omega)$
satisfies the equation

$\frac{1}{2\pi i}\oint\omega\frac{d}{d\omega}\ln
g(\omega)d\omega=\sum\omega_{0}-\sum\omega_{\rm poles},$ (48)

where $\omega_{0}$ are the zeros and $\omega_{\rm poles}$ are the poles of
$g(\omega)$ inside the contour of integration. This contour is taken to be a
semicircle of large radius $R$ in the right half plane. A definite advantage
of this method is that the multiplicities of zeros and poles are automatically
included. The argument principle was introduced in connection with the Casimir
effect by van Kampen et al. [45], and has later been made use of extensively
(cf., for instance, Refs. [8, 5, 8, 46]).

For the function $g(\omega)$ it is natural to start from the expression for
$D$ in Eq. (36), but we have to normalize it in a convenient way. We first
introduce a convenient parametrization which relates the pieces $L_{I}$ and
$L_{II}$ to the total length $L$,

$L_{I}=pL,\quad L_{II}=qL,\quad p+q=1.$ (49)

Then, define the quantity $A$ as

$A=\frac{1}{4}(1+x)^{2}\cos[(n_{1}p+n_{2}q)\omega
L]-\frac{1}{4}(1-x)^{2}\cos[(n_{1}p-n_{2}q)\omega L].$ (50)

For a uniform string, $x=1~{}(n_{1}=n_{2}=n)$, we have $A=\cos(n\omega L)$. We
now define $g(\omega)$ as

$g(\omega)=\left|\frac{1-A}{A}\right|.$ (51)

With this form, the big semicircle does not contribute to the integration. Let
$\omega=i\xi$, where $\xi$ is the frequency along the imaginary axis. This
Wick rotation implies that the quantity $A$ goes into

$A\rightarrow A(\xi)=\frac{1}{4}(1+x)^{2}\cosh[(n_{1}p+n_{2}q)\xi
L]-\frac{1}{4}(1-x)^{2}\cosh[(n_{1}p-n_{2}q)\xi L].$ (52)

By performing a partial integration along the imaginary axis, and observing
that the positive and negative frequencies contribute equally, we obtain for
the Casimir energy

$E=\frac{1}{2\pi}\int_{0}^{\infty}\ln\left|\frac{1-A(\xi)}{A(\xi)}\right|d\xi.$
(53)

With our adopted normalization, we can now test the results for $E$ calculated
in special cases. To simplify the formalism somewhat, we choose $n_{1}=1$, the
lowest possible value for $n_{1}$ in nondispersive theory. Then $n_{2}=1/x$.
Let us also choose $L=1$. This implies that the expression for $A(\xi)$
becomes simpler,

$A(\xi)=\frac{1}{4}(1+x)^{2}\cosh\left[\left(p+\frac{q}{x}\right)\xi\right]-\frac{1}{4}(1-x)^{2}\cosh\left[\left(p-\frac{q}{x}\right)\xi\right].$
(54)

1\. The case of a uniform string.

In this case $A(\xi)\big{|}_{x=1}=\cosh\xi$ for all $p$, and we get

$E_{\rm
uniform}=\frac{1}{2\pi}\int_{0}^{\infty}\ln\left|\frac{1-\cosh\xi}{\cosh\xi}\right|d\xi.$
(55)

The integral can be calculated analytically by means of polylogarithm
functions and is equal to

$E_{\rm uniform}=-3\frac{\pi}{16}.$ (56)

It is worth noticing it is different from (43) by a multiplicative factor
$\frac{9}{8}$. One could think that this is an inconsistency as the
calculations seem to refer to the same Casimir energy $E$ for the homogenous
string. Actually, the two situations are different:
The energy of Eq. (43) is calculated for a scalar field, so quantum numbers
run from $N=1$ to infinity , while Eq. (56) is calculated for the
electromagnetic field, where $N=0$ contributions are not zero. It is known in
electromagnetic Casimir problems for dielectrics that both TE and TM
polarizations contribute with zero frequency [47]. Although the two cases are
not directly comparable, we note that the result of the integral (56) is given
from

$E_{\rm
uniform}=-\frac{1}{2\pi}(4\operatorname{Li}_{2}(1)-\operatorname{Li}_{2}(-1)),$
(57)

where the first term in brackets gives the energy (43), and the second term
gives the difference.

2\. The case when $p=1/2.$

The string is then divided into two equal halves, the refractive index ratio
$x$ being arbitrary. Now

$A(\xi)\big{|}_{p=1/2}=\frac{1}{4}(1+x)^{2}\cosh\left[\left(1+\frac{1}{x}\right)\frac{\xi}{2}\right]-\frac{1}{4}(1-x)^{2}\cosh\left[\left(1-\frac{1}{x}\right)\frac{\xi}{2}\right],$
(58)

and the Casimir energy is found by inserting this expression into Eq. (53). It
is not trivial to find an analytical solution for the integral (53), so we
calculated it numerically by a Python program. We show the resulting plot in
Fig. 2.
It is worth noticing how the energy $E$ is positive or negative depending on
the value of $x$. For $x\lessapprox 0.15$, $E$ is positive and has a maximum
at $x\approx 0.04$; for larger $x$ the energy is negative, until it reaches
the value (56) for $x=1$.

Figure 2: Graph of $E$ versus $x$ in the case 2, where $x$ is restricted to
the region $0<x<1$. We notice how $E$ can be positive or negative depending on
the value of $x$: for $x\lessapprox 0.15$, $E$ is positive and has a maximum
at $x\approx 0.04$; for larger $x$ the energy is negative until it reaches the
value (56) for $x=1$.

3\. The case when $p\rightarrow 0$.

This case is of interest since it corresponds to a ”particle” sitting on a
uniform string. We have now $A(\xi)\big{|}_{p\rightarrow 0}=x\cosh(\xi/x)$,
and the Casimir energy becomes

$E=\frac{1}{2\pi}\int_{0}^{\infty}\ln\left|\frac{1-x\cosh(\xi/x)}{x\cosh(\xi/x)}\right|d\xi.$
(59)

As before, we assume that $x$ lies in the interval $0<x<1$. Figure 3 shows the
same behavior except from the middle region $0.15\lessapprox x<1$, where the
energy of case 3 diminishes slower than in case 2.
These similarities in the extremes for $x\lessapprox 0.15$ and $x=1$ can be
understood. It is easy to verify that the integrals into Eq. (58) and Eq. (59)
give the same result for $x=0$, because we have no fields due to boundary
conditions and the medium in region II behaving as a perfect mirror so we get
$E=0$. The same is valid for $x=1$, as the cases 1,2 and 3 are equivalent when
the string is uniform.

Figure 3: Graph of $E$ versus $x$ in the case 3, analogous to Fig. 2. We
notice how $E$ is positive or negative depending on the value of $x$: for
$x\lessapprox 0.15$, $E$ is positive and has a maximum at $x\lessapprox 0.04$;
for larger $x$, the energy is negative until it reaches Eq. (56) for $x=1$.
The main difference with Fig. 2 relies on the behaviour of the function in the
middle region, where the energy of case 3 diminishes slower than in the case
2.

##  VI Curvature-induced enhancement of the produced electric field at a
dielectric boundary

As our last point we will return to the axion field, with now the purpose of
focusing on a characteristic property of the extra produced electric field
$E^{\gamma}$ occurring in cylindrical geometry such as a haloscope. A
considerable enhancement of this field appears to occur near the centre of the
cylinder. The enhancement is solely caused by the curvilinear geometry. To our
knowledge this effect has not been considered before.

Assume there is an infinitely long cylindrical vacuum region with radius $R$,
surrounded by an exterior massive nonmagnetic environment. As before, a strong
uniform static magnetic field ${\bf B}_{e}$ is acting in the $z$ direction. We
assume there are only the time-varying axions present in the system, together
with the extra photons $E^{\gamma}$ that they generate at the boundary. The
solutions of the Bessel equations in the exterior and interior regions are
cylinder functions. No propagation of electromagnetic modes are assumed to
take place in the $z$ direction, and we can start from the usual
electromagnetic mode expressions for the axial electric and the azimuthal
magnetic field components in the exterior and interior regions (cf., Ref.
[44], page 525)

$E_{z}^{\rm ext}=H_{0}^{(1)}(k_{2}r)a_{\rm ext}e^{-i\omega t},\quad
H_{\theta}^{\rm ext}=in_{2}H_{0}^{(1)^{\prime}}(k_{2}r)a_{\rm ext}e^{-i\omega
t},$ (60) $E_{z}^{\rm int}=J_{0}(\omega r)a_{\rm int}e^{-i\omega t},\quad
H_{\theta}^{\rm int}=iJ_{0}^{\prime}(\omega r)a_{\rm int}e^{-i\omega t}.$ (61)

Here $H_{p}^{(1)}$ is the Hankel functions of the first kind, of order $p$.
Azimuthal symmetry is assumed, so that only $p=0$ applies. Outgoing waves are
assumed on the outside, and on the inside a stationary wave field is assumed,
finite at the center.

In the present case the situation is different, as the antenna-like property
of the boundary $r=R$ causes radiation to occur into the inward also. That
means, $J_{0}$ has to be replaced by the Hankel function $H_{0}^{(2)}$ of the
second kind. Thus Eq. (61) gets replaced by

$E_{z}^{\rm int}=H_{0}^{(2)}(\omega r)a_{\rm int}e^{-i\omega t},\quad
H_{\theta}^{\rm int}=iH_{0}^{(2)^{\prime}}(\omega r)a_{\rm int}e^{-i\omega
t}.$ (62)

As before, $k_{2}=n_{2}\omega$ with $n_{2}$ real. The coefficients $a_{\rm
ext}$ and $a_{\rm int}$ are determined by the boundary conditions at $r=R$.
Looking back at the treatment in Sec. III, we will require the correspondence

$E_{2}^{\gamma}\rightarrow H_{0}^{(1)}(k_{2}R)a_{\rm ext}e^{-i\omega t},\quad
E_{1}^{\gamma}\rightarrow H_{0}^{(2)}(\omega R)a_{\rm int}e^{-i\omega t}.$
(63)

To simplify the formalism somewhat, we assume in the following that the radius
$R$ is so large that one may use the approximate versions for the Hankel
functions for large arguments at the boundary. As is known,

$H_{0}^{(1)}(\rho)=\sqrt{\frac{2}{\pi\rho}}\,e^{i(\rho-\pi/4)},\quad
H_{0}^{(2)}(\rho)=\sqrt{\frac{2}{\pi\rho}}\,e^{-i(\rho-\pi/4)},\quad\rho\gg
1.$ (64)

As $H_{0}^{(1)^{\prime}}(\rho)=iH_{0}^{(1)}(\rho)$,
$H_{0}^{(2)^{\prime}}(\rho)=-iH_{0}^{(2)}(\rho)$, we can write the boundary
conditions as

$H_{0}^{(1)}(k_{2}R)a_{\rm
ext}-\frac{E_{0}}{\varepsilon_{2}}=H_{0}^{(2)}(\omega R)a_{\rm int}-E_{0},$
(65) $n_{2}H_{0}^{(1)}(k_{2}R)a_{\rm ext}=-H_{0}^{(2)}(\omega R)a_{\rm int},$
(66)

where $E_{0}=\theta_{0}B_{e}$, as before. The coefficients $a_{\rm ext}$ and
$a_{\rm int}$ can then be found, and we can express the result as

$H_{0}^{(1)}(k_{2}R)a_{\rm
ext}=-\frac{E_{0}}{n_{2}}\left(1-\frac{1}{n_{2}}\right),$ (67)
$H_{0}^{(2)}(\omega R)a_{\rm int}=E_{0}\left(1-\frac{1}{n_{2}}\right),$ (68)

in agreement with Eq. (21) obtained for plane geometry.

We now move on to the central point in this analysis, which is to observe the
behavior of the inward-emitted electric field $E_{z}$ near the center of the
cylinder. As the magnitude of $H_{0}^{(2)}(\rho)$ increases logarithmically as
$\rho\rightarrow 0$, there will necessarily be an enhancement of the field in
this region. Now, one might argue that the accumulation of electric fields
near $r=0$ would lead to an unstable situation. We may avoid this, at least
conceptually, by assuming there to be a perfect cylindrical absorber of small
radius $r=\delta$ centered at the $z$ axis, thus allowing for stable
conditions.

Letting the inward-generated field at $r=R$ be represented by
$H_{0}^{(2)}(\omega R)$, we may for small values of the argument $\rho$ make
use of the approximation

$H_{0}^{(2)}(\rho)=J_{0}(\rho)-iN_{0}(\rho)=1+\frac{2i}{\pi}\ln\frac{2}{\gamma\rho},\quad\rho\ll
1,$ (69)

with $\gamma=1.78107$. We need only consider the magnitudes of the field
expressions. Thus the magnitudes of the fields measured at the minimum radius
$\delta$ and at the cylinder radius $R$ are related by

$\left|\frac{E_{z}^{\gamma}(\delta)}{E_{z}^{\gamma}(R)}\right|=\left|\frac{H_{0}^{(2)}(\omega\delta/c)}{H_{0}^{(2)}(\omega
R/c)}\right|=\left|\frac{1+\frac{2i}{\pi}\ln\frac{2c}{\gamma\omega\delta}}{H_{0}^{(2)}(\omega
R/c)}\right|$ (70)

in physical units.

We make finally some numerical estimates. Take $R=20~{}$cm as a reasonable
value for the cylinder radius, and choose for definiteness
$m_{a}c^{2}=10^{-4}~{}$eV for the axion energy, what corresponds to
$\omega=1.52\times 10^{11}~{}$rad/s. Then $\omega R/c=1.0\times 10^{2}$, thus
justifying use of the approximation (64). We get $|H_{0}^{(2)}(\omega
R/c)|=0.080~{}$. With the minimum radius chosen as $\delta=100~{}\mu$m,
corresponding to $\rho_{\rm min}=\omega\delta/c=0.051$, it is seen that also
the low-argument approximation (69) is justified. We obtain for the ratio (70)

$\left|\frac{E_{z}^{\gamma}(\delta)}{E_{z}^{\gamma}(R)}\right|=12.5\times|1+2.0\,i|=28.0.$
(71)

There occurs thus a considerable enhancement of the amplitude in the
neighbourhood of the cylinder center. While the numerical values above were
chosen so as to satisfy the mathematical approximations, there is clearly a
flexibility of choosing different parameters that can better fit experimental
conditions.

We emphasize again that this enhancement is solely a geometrical focusing
effect, being an extension of the theory worked out earlier [1] for plane
geometry. An observation of the increased signal near $r=0$ may be of
experimental interest. The idea may be looked upon as an alternative to the
idea recently put forward in Ref. [2], concerning the broadband solenoidal
haloscope.

## Acknowledgments

We thank the Erasmus project, the Norwegian University of Science and
Technology and the University of Palermo for the support of the Erasmus
Traineeship of the student Amedeo Maria Favitta, during which part of this
work was performed in the context of his work for the Master Thesis.

## References

  * [1] A. J. Millar, G. R. Raffelt, J. Redondo and F. D. Steffen, J. Cosm. Astropart. Phys. 01(2017) 061; arXiv:1612.07057.
  * [2] J. Liu, K. Dona, G. Hoshino, S. Knirck. N. Kurinsky, N. Malaker et al., Phys. Rev. Lett. 128, 131801 (2022).
  * [3] I. Brevik and H. B. Nielsen, Phys. Rev. 41, 1185 (1990).
  * [4] X. Li, X. Shi and J. Zhang, Phys. Rev. D 44, 560 (1991).
  * [5] I. Brevik and E. Elizalde, Phys. Rev. D 49, 5319 (1994).
  * [6] I. Brevik, H. B. Nielsen and S. D. Odintsov, Phys. Rev. D 53, 3224 (1996).
  * [7] I. Brevik, E. Elizalde, R. Sollie and J. B. Aarseth, J. Math Phys. 40, 1127 (1999).
  * [8] I. Brevik, A. A. Bytsenko and B. M. Pimentel, ’Thermodynamic properties of the relativistic composite string - expository remarks’, in Theoretical Physics 2002, Part 2 (Nova Science Publ. 2003), p. 117.
  * [9] M. Bordag and I. G. Pirozhenko, arXiv:2012.14301 [hep-th].
  * [10] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
  * [11] R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977).
  * [12] P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983).
  * [13] S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
  * [14] J. Preskill, M. B. Wise and F. Wilczek, Phys. Lett. B 120, 127 (1983).
  * [15] L. F. Abbott and P. Sikivie, Phys. Lett. B 120, 133 (1983).
  * [16] M. Dine and W. Fischler, Phys. Lett. B 120, 137 (1983).
  * [17] P. Sikivie, ’Axion cosmology’, in Springer Lecture Notes in Physics 741 (2008), edited by M. Kuster, G. Raffelt and B. Beltran, pp. 19-50.
  * [18] P. Sikivie, N. Sullivan and D. B. Tanner, Phys. Rev. Lett. 112, 131301 (2014).
  * [19] M. Lawson, A. J. Millar, M. Pancaldi, E. Vitagliano, and F. Wilczek, Phys. Rev. Lett. 123, 141802 (2019); arXiv:1904.11872 [hep-ph].
  * [20] Y. Kim, D. Kim, J. Jeong, J. Kim, Y. C. Shin and Y. K. Semertzidis, Phys. Dark Universe 26, 100362 (2019).
  * [21] Q.-D. Jiang and F. Wilczek, Phys. Rev. B 99, 125403 (2019); arXiv:1805.07994 [cond-mat.mes-hall].
  * [22] P. Sikivie, Rev. Mod. Phys. 93, 15004 (2021); arXiv:2003.02206 [hep-ph].
  * [23] J. I. McDonald and L. B. Ventura, Phys. Rev. D 101, 123503 (2020); arXiv:1911.10221 [hep-ph].
  * [24] M. Chaichian, I. Brevik and M. Oksanen, 40th Int. Conf. on High Energy Phys. (ICHEP 2020) Proceedings of Science, Vol. 390; arXiv:2101.05758 [astro-ph.HE].
  * [25] P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * [26] A. Arza, T, Schwetz and E. Todarello, arXiv:2004.01669v2 [hep-ph].
  * [27] P. Carenza, A. Mirizzi and G. Sigl, Phys. Rev. D 101, 103016 (2020).
  * [28] M. Leroy, M. Chianese, T. D. P. Edwards and C. Weniger, Phys. Rev. D 101, 123003 (2020).
  * [29] I. Brevik, M. Chaichian and M. Oksanen, Eur. Phys. J. C 81, 926 (2021); arXiv:2101.00954 [astro-ph.HE].
  * [30] I. Brevik, Universe 7, 133 (2021); arXiv:2202.11152 [hep-ph].
  * [31] J. Ouellet and Z. Bogorad, Phys. Rev. D 99, 055010 (2019).
  * [32] A. Arza and P. Sikivie, Phys. Rev. Lett. 123, 131804 (2019); arXiv:1902.00114 [hep-ph].
  * [33] Z. Qiu, G. Cao, and X.-G. Huang, Phys. Rev. D 95, 036002 (2017).
  * [34] J. A. Dror, H. Murayama, and N. L. Rodd, Phys. Rev. D 103, 115004 (2021).
  * [35] I. Brevik and M. Chaichian, Eur. Phys. J. C 82, 202 (2022); arXiv:2202.09882 [hep-ph].
  * [36] K. Fukushima, S. Imaki and Z. Qiu, Pys. Rev. D 100, 045013 (2019).
  * [37] M. E. Tobar, B. T. McAllister and M. Goryachev, Phys. Dark Universe 26, 100339 (2019); arXiv:1809.01654 [hep-ph].
  * [38] S. Bae, SungWoo Youn and J. Jeong, arXiv:2205.08885 [hep-ex].
  * [39] P. Adshead, P. Draper and B. Lillard, Phys. Rev. D 102, 123011 (2020).
  * [40] A. Patkos, Symmetry 14, 1113 (2022).
  * [41] M. E. Tobar, B. T. McAllister and M. Goryachev, Phys. Rev. D 105, 045009 (2022).
  * [42] W. DeRocco and A. Hook, Phys. Rev. D 98, 035021 (2018).
  * [43] I. H. Brevik and M. M. Chaichian, Int. J. Mod. Phys. A 2250151 (2022); arXiv:2207.04807 [hep-ph].
  * [44] J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, 1941).
  * [45] N. G. van Kampen, B. R. A. Nijboer and K. Schram, Phys. Lett. A 26, 307 1968).
  * [46] Yu. S. Barash and V. L. Ginzburg, in ”Electromagnetic fluctuations and molecular forces”, edited by L. V. Keldysh, D. A. Kirzhnitz and A. A. Maradudin, Chap. 6 (Elsevier, Amsterdam, 1989).
  * [47] K. A. Milton, The Casimir effect, Physical Manifestations of Zero-Point Energy (World Scientific, Singapore, 2001).