# Wave functions for high-symmetry, thin microstrip antennas and two-
dimensional quantum boxes

Joseph R. Rain  Department of Physics, University of Central Florida, 4111
Libra Drive, Orlando, Florida, 32816-2385, USA  PeiYu Cai  Department of
Physics, University of Central Florida, 4111 Libra Drive, Orlando, Florida,
32816-2385, USA  University College London, Department of Physics, South
Kensington Campus, London SW7 2AZ, UK*  Alexander Baekey  Department of
Physics, University of Central Florida, 4111 Libra Drive, Orlando, Florida,
32816-2385, USA  Matthew A. Reinhard  Department of Physics, University of
Central Florida, 4111 Libra Drive, Orlando, Florida, 32816-2385, USA
Department of Physics, University of Florida, 2001 Museum Road, P. O. Box
118440, Gainesville, Florida, 32611-8440, USA*  Roman I. Vasquez  Department
of Physics, University of Central Florida, 4111 Libra Drive, Orlando, Florida,
32816-2385, USA  Department of Mathematics and Statistics, Auburn University,
221 Roosevelt Concourse, Auburn, Alabama 36849, USA*  Andrew C. Silverman
Department of Physics, University of Central Florida, 4111 Libra Drive,
Orlando, Florida, 32816-2385, USA  Christopher L. Cain  Department of Physics,
University of Central Florida, 4111 Libra Drive, Orlando, Florida, 32816-2385,
USA  Engineering Technology, Mechatronics, St. Petersburg College, Clearwater,
Florida 37765, USA*  Richard A. Klemm† Department of Physics, University of
Central Florida, 4111 Libra Drive, Orlando, Florida, 32816-2385, USA  U. S.
Research Laboratory, Wright-Patterson Air Force Base, Ohio, 45433-7251, USA

###### Abstract

For a spinless quantum particle in a one-dimensional box or an electromagnetic
wave in a one-dimensional cavity, the respective Dirichlet and Neumann
boundary conditions both lead to non-degenerate wave functions. However, in
two spatial dimensions, the symmetry of the box or microstrip antenna is an
important feature that has often been overlooked in the literature. In the
high-symmetry cases of a disk, square, or equilateral triangle, the wave
functions for each of those two boundary conditions are grouped into two
distinct classes, which are one- and two-dimensional representations of the
respective point groups, $C_{\infty v}$, $C_{4v}$, and $C_{3v}$. Here we
present visualizations of representative wave functions for both boundary
conditions and both one- and two-dimensional representations of those point
groups. For the one-dimensional representations, color contour plots of the
wave functions are presented. For the two-dimensional representations, the
infinite degeneracies are presented as common nodal points and/or lines, the
patterns of which are invariant under all operations of the respective point
group. The wave functions with the Neumann boundary conditions have important
consequences for the coherent terahertz emission from the intrinsic Josephson
junctions in the high-temperature superconductor Bi2Sr2CaCu2O8+δ: the
enhancement of the output power from electromagnetic cavity resonances is only
strong for wave functions that are not degenerate.

*present address †corresponding author. email<EMAIL_ADDRESS>
##  I Introduction

The study of wave functions obtained from various geometries with Dirichlet
and Neumann boundary conditions has been a useful educational resource and
also has numerous applications in the construction of various devices. A
quantum particle in a one-dimensional infinite square well potential, or
“box”, for which the boundary is a set of two points, is often the first
problem studied by undergraduate students early in their first course on
quantum mechanics GS . Although two graduate texts contain two problems on the
degeneracies of the lowest energies of a square two-dimensional (2D) box
Sakurai ; SN , the solutions manuals for those texts both give an incorrect
value for the degeneracy of the square box’s first excited state. Hence, there
could be considerable confusion on this issue. Here we focus upon high-
symmetry, 2D shapes. For a spinless quantum particle of mass $M$ in a 2D
infinite square well potential or box, the wave function $\psi(x,y,t)$
satisfies the Schrödinger equation,

$\displaystyle-\frac{\hbar^{2}}{2M}{\bm{\nabla}}^{2}\psi+V\psi$
$\displaystyle=$ $\displaystyle i\hbar\frac{\partial\psi}{\partial t},$ (1)

where $\hbar=\frac{h}{2\pi}$ and $h$ is Planck’s constant, for which the
potential $V(x,y)=0$ inside the box and $V(x,y)=\infty$ outside it. Hence
$\psi(x,y,t)=0$ outside the box and on its boundary, the simplest example of
Dirichlet boundary conditions. Here we only consider closed one-dimensional
boundaries, and focus upon the 2D shapes with the highest point-group
symmetries, $C_{\infty v}$, $C_{4v}$, and $C_{3v}$, corresponding to
cylindrical, square, or equilateral triangular boxes Tinkham .

For a thin (nearly 2D) microstrip antenna (MSA), the magnetic vector potential
$A_{z}(x,y,t)$ normal to the antenna satisfies the electromagnetic (EM) wave
equation

$\displaystyle{\bm{\nabla}}^{2}A_{z}-\frac{1}{v^{2}}\frac{\partial^{2}A_{z}}{\partial
t^{2}}$ $\displaystyle=$ $\displaystyle 0,$ (2)

where $v$ is the wave velocity that depends upon the index of refraction in
the antenna, and for transverse magnetic (TM) modes, its normal derivative
vanishes on the boundary, the simplest example of Neumann boundary conditions.
Due to the oscillatory time dependence of a light wave, Eq. (2) is usually
rewritten as ${\bm{\nabla}}^{2}A_{z}+(k^{\prime})^{2}A_{z}=0$, where
$k^{\prime}$ is the wave vector in the material of interest.

In particular, determining the symmetries and energy states of wave functions
with Neumann boundary conditions is of great practical importance in the
development of a high-power terahertz (THz) laser, which has many potential
applications, such as for the detection of skin or colon cancer in humans and
in secure communications. This is due to the ac Josephson effect, in which a
dc voltage $V$ is applied across a single junction, leading to an ac current
$I$ and the emission of photons at the frequency $f_{J}=2eV/h$, where $e$ is
the charge of an electron Josephson . Now there exist many layered
superconductors Klemmbook , a number of which exhibit Josephson effects, but
the most interesting one for the construction of a THz laser is the high
transition temperature $T_{c}$ superconductor Bi2Sr2CaCu2O8+δ (Bi2212). This
material consists of a uniform stack of intrinsic Josephson junctions (IJJs)
Kleiner1992 ; Kleiner1994 ; Batov2006 ; Ozyuzer2007 . The output power $P_{1}$
of a single IJJ is about 1 pW, too small for most practical applications that
require the actual power $P$ to be at least 1 mW. But since each IJJ is 1.533
nm thick, a single crystal of Bi2212 of thickness 1 $\mu$m contains $N\approx
650$ IJJs, reducing $f_{J}$ to

$\displaystyle f_{J}$ $\displaystyle=$ $\displaystyle(2e/h)(V/N),$ (3)

and when most of the $N$ junctions emit coherently, ideally,
$P_{N}=P_{1}N^{2}\approx 0.4\mu$W. Moreover, in a thin mesa cut from a single
crystal of Bi2212, the shape of the mesa acts as an electromagnetic cavity or
MSA, which can enhance the output power an additional one to two orders of
magnitude Ozyuzer2007 ; Kadowaki2010 ; Klemm2010a ; Klemm2010b ; Tsujimoto2012
. The emission frequency has to be larger than the Josephson plasma resonance
frequency $f_{p}\approx 0.25$ THz Singley2004 . In principle, the maximum
emission frequency is the low temperature $T$ value of the superconducting gap
$2\Delta$, which is about 15 THz Hoogenboom ; Borodianskyi2017 ; Xue2016 ;
Xue2020 .

However, a major issue affecting the reliability and the upper limit of the
emission frequency has been the Joule heating of the mesasWang2009 ; Wang2010
; Minami2014 , especially when they have Bi2212 substrates, since Bi2212 is a
very poor thermal conductor. But this problem has been mostly removed by
fabricating stand-alone mesas, in which the Bi2212 sample is doubly cleaved to
a thickness of 1-2 microns from a single crystal, and the top and bottom
surfaces are each coated with a thin layer of gold Klemm2010a ; Klemm2010b ;
Tsujimoto2012 ; Klemm2010c ; Kitamura2014 ; Kashiwagi2015b ; Kashiwagi2018 ;
Kashiwagi2017b . These issues are discussed in Section VIII. When different
parts of a small stand-alone Bi2212 single crystal were used both as the
emitter and as the detector, emission up to 11 THz was observed
Borodianskyi2017 . An array of three stand-alone rectangular mesas was
reported to emit coherently with the combined $P\approx 0.61$ mW Benseman2013
, although an array of two rectangular mesas did not behave coherently
Kleiner2015 , so the main concern for further development is the design of
arrays to increase that number.

For low-symmetry rectangular boxes or MSAs, the wave functions are all
nondegenerate, one-dimensional representations (1DRs) of the point group
$C_{2v}$. However, for the higher symmetry square Klemm2017 ; Klemm2017b ;
Sun2018 ; Ono2020 , equilateral triangular Helszajn1978 ; Overfelt1986 ;
Delfanazari2012 ; Delfanazari2013 ; Delfanazari2013b ; Klemm2013 ;
Cerkoney2017 , isosceles and right triangular Delfanazari2012b ;
Delfanazari2014 , regular pentagonal Delfanazari2015 , cylindrical, disk
Tsujimoto2010 ; Kashiwagi2015 , annular Bonnough2018 , or singly slitted
annularShouk2021  shapes, the situation can be considerably more complicated.
Although some of the wave functions are non-degenerate 1DRs of the respective
point groups $C_{\infty v}$, $C_{4v}$ and $C_{3v}$ under consideration here, a
large fraction of the wave functions are infinitely-degenerate two-dimensional
representations (2DRs) of those point groups Tinkham . In addition, for square
boxes or MSAs, many additional wave functions are doubly degenerate. Previous
works have calculated the dimensionality of the symmetry groups for the
stationary states of the various wave functions, but there have not been
thorough investigations of the features of the 2DRs of all three of these
cases. Although for 2D boxes, such considerations are only experimentally
relevant for deep quantum wells, for nearly 2D MSAs, the experimental
consequences are very important, but have not been clearly described in the
literature.

We note that circularly polarized coherent THz emission can be obtained by
breaking the symmetry of square or disk MSAs Asai2017 ; Elarabi2017 ;
Elarabi2018 , Bi2212 MSAs can be used both as emitters and as detectors
Borodianskyi2017 ; Irie2012 , and commercial cryocoolers can be used in
cooling Bi2212 MSAs for many potential applications Nakade2014 ; Saiwai2020 .
To date, six review articles on Bi2212 IJJ-THz emitters have been published
Kashiwagi2012 ; Welp2013 ; Kakeya2016 ; Kashiwagi2017 ; Kleiner2019 ;
Delfanazari2020 .

Here we present detailed studies of the wave functions for the three highest-
symmetry 2D shapes: cylindrical boxes and disk MSAs, square boxes and MSAs,
and equilateral triangular boxes and MSAs, in which either the wave function
or its normal derivative vanishes on the boundary. Character tables of the
respective point groups $C_{\infty v}$, $C_{4v}$ and $C_{3v}$ are given in
textbooks on group theory Tinkham , but there are some minor differences in
the wave function tables with Dirichlet and Neumann boundary conditions we
present here, and some additions for the 2DR wave functions. In Section II, we
analyze the square box. In Section III, we describe the thin square MSA. In
Section IV, we analyze the equilateral triangular box. In Section V, we
present the results for thin equilateral triangular MSA wave functions. In
Section VI, we show the cylindrical box wave functions. In Section VII, we
describe the disk MSA wave functions. In Section VIII, we compare our results
for square, equilateral triangular, and disk MSAs with published experimental
results. Finally in Section IX, we summarize our results for these high-
symmetry boxes and MSAs.

##  II The square box

For a spinless quantum particle of mass $M$ in a square box of side $a$, the
normalized wave functions are solutions of the Schrödinger equation with
$V(x,y)=0$ for $0<x,y<a$, $V(x,y)=\infty$ for $x,y\leq 0$ and $x,y\geq a$,
which are

$\displaystyle\Psi_{n,m}(x,y)$ $\displaystyle=$
$\displaystyle\frac{2}{a}\sin(n\pi x/a)\sin(m\pi y/a),$ (4)

for integral $n,m\geq 1$, all of which satisfy the Dirichlet boundary
conditions $\Psi_{n,m}(x,y)=0$ for $x=0,a$ and $y=0,a$. The energy of that
state is

$\displaystyle E_{n,m}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}(n^{2}+m^{2})\pi^{2}}{2Ma^{2}}.$ (5)

Figure 1 displays color-coded plots of the wave functions $\Psi_{n,m}(x,y)$
for $1\leq n,m\leq 4$ of the square box.

Figure 1: (color online) Table of color-coded plots of the normalized wave
functions $\Psi_{n,m}(x,y)=\frac{2}{a}\sin(n\pi x/a)\sin(m\pi y/a)$ for a
square box of side $a$ with Dirichlet boundary conditions for $(n,m)=1,2,3,4$,
each with its lower left corner at (0,0). Here we set $a=1$. $n$ and $m$ are
respectively the column and row numbers. The color code bar applies to each of
these figures. The black boundaries and internal lines are nodes.

According to the $C_{4v}$ point group symmetry class, there are four mirror
planes: the horizontal $\sigma_{h}$ and vertical $\sigma_{v}$ mirror planes
that bisect the sides, and the two diagonal mirror planes $\sigma_{d1}$ and
$\sigma_{d2}$ that bisect the corners. In addition, two rotations ($R_{4}$) by
$2\pi/4$ and one ($R_{2}$) by $\pi$ about the centroid are also allowed
Tinkham . The wave functions fall into three basic classes. In the first
class, $n=m$, $\Psi_{n,n}(x,y)$ is nondegenerate. But there are two subclasses
of these nondegenerate wave functions. For $n$ odd, the $\Psi_{n,n}(x,y)$ are
invariant under all of these operations, whereas for $n$ even, the
$\Psi_{n,n}(x,y)$ are even about $\sigma_{d1},\sigma_{d2}$ and under $R_{2}$,
but are odd about $\sigma_{h},\sigma_{v}$ and under $R_{4}$. According to
Table I, the odd and even $n$ $\Psi_{n,n}(x,y)$ are respectively elements of
subgroups $A_{1}$ and $B_{2}$.

Figure 2: (color online)(a) Color-coded plots of the infinitely-degenerate
square box wave function
$\Psi^{-\theta,+}_{1,2}(x,y)=\cos\theta\Psi_{1,2}(x,y)-\sin\theta\Psi_{2,1}(x,y)$
for $\phi=0$, $\theta=30^{\circ}$ (left), $45^{\circ}$ (center), and
$60^{\circ}$ (right). (b) Color-coded illustration of the oddness of
$\Psi_{2,1}(x,y)$ under rotations about its centroid at
$(\frac{a}{2},\frac{a}{2})$ by $\pi$.

For $m=n+2p+1$, the pair of wave functions $\Psi_{n,n+2p+1}(x,y)$ and
$\Psi_{n+2p+1,n}$ are odd under $R_{2}$, and have $E=2$, the trace of the rank
2 identity matrix ${\bm{1}}$. The full degeneracy may be represented by

$\displaystyle\Psi^{(\theta,+)}_{n,n+2p+1}$ $\displaystyle=$
$\displaystyle\cos\theta\Psi_{n,n+2p+1}(x,y)$
$\displaystyle+e^{i\phi}\sin\theta\Psi_{n+2p+1,n}(x,y),$
$\displaystyle\Psi^{(\theta,-)}_{n,n+2p+1}$ $\displaystyle=$
$\displaystyle-\sin\theta\Psi_{n,n+2p+1}(x,y)$ (6)
$\displaystyle+e^{i\phi}\cos\theta\Psi_{n+2p+1,n}(x,y),$

which are the two orthonormal subsets of the degenerate $\Psi_{n,n+2p+1}$ and
$\Psi_{n+2p+1,n}$ wave functions Klemm2017b . However, since
$0\leq\theta<2\pi$, $\theta$ can be any real mixing angle, this degeneracy is
infinite. For simplicity, we shall assume that $\phi=0$, so that the wave
functions are all real. This is entirely analogous to a spin $\frac{1}{2}$
system in the absence of a magnetic field SN . In Fig. 2(a), this degeneracy
is displayed for $\Psi_{1,2}^{-\theta,+}(x,y)$ for $\phi=0$ at the three
mixing angles $\theta=30^{\circ},45^{\circ}$, and $60^{\circ}$. In Fig. 2(b),
the fact that such wave functions are odd under $R_{2}$ is evident for
$\Psi_{2,1}(x,y)$ by adding $\Psi_{2,1}(x)+R_{2}\Psi_{2,1}(x,y)$, which
vanishes. The only points at which such wave functions are invariant under all
of the operations of $C_{4v}$ are their common nodal points. These wave
functions are thus displayed for $1\leq n,m\leq 4$ as the appropriate sets of
nodal points in Fig. 3. We note that in each of these cases, the complete set
of nodal points is invariant under each of the operations of $C_{4v}$ (all
four mirror planes and both rotations).

Writing these 2DR wave functions in the Nambu representation,

$\displaystyle\Psi^{\theta}_{n,n+2p+1}(x,y)$ $\displaystyle=$
$\displaystyle\Biggl{(}\begin{array}[]{c}\Psi^{(\theta,+)}_{n,n+2p+1}(x,y)\\\
\Psi^{(\theta,-)}_{n,n+2p+1}(x,y)\end{array}\Biggr{)},$ (9)

where the $\Psi^{(\theta,\pm)}_{n,n+2p+1}(x,y)$ are given by Eq. (6), the
operations of $C_{4v}$ upon them are rank-2 matrices. It is easy to show that
$R_{2}=-{\bm{1}}$, the trace of which is -2. The other operations can be
written in terms of the Pauli matrices $\sigma_{x}$, $\sigma_{y}$, and
$\sigma_{z}$. In particular, the 2 $R_{4}$ matrices are $\pm i\sigma_{y}$,
$\sigma_{h}=\sigma_{v}=\sigma_{z}\cos(2\theta)-\sigma_{x}\sin(2\theta)$, and
$\sigma_{d1}=-\sigma_{d_{2}}=\sigma_{z}\sin(2\theta)+\sigma_{x}\cos(2\theta)$.
Since the Pauli matrices are traceless, the traces of the 2 $R_{4}$,
$\sigma_{v}$, $\sigma_{h}$, $\sigma_{d1}$ and $\sigma_{d2}$ all vanish for any
value of $\theta$.

Figure 3: (color online) Table of color-coded plots of the square box non-degenerate $\Psi_{n,n}(x,y)$ along the table diagonal, the doubly-degenerate $\Psi^{+}_{n,n+2p}(x,y)$ and $\Psi^{-}_{n,n+2p}(x,y)$ are displayed in the regions above and below the diagonal, respectively, and the infinitely degenerate $\Psi_{n,n+2p+1}^{\theta,\pm}(x,y)$ are represented by their common nodal points. Each such set of common nodal points is a one-dimensional representation of $C_{4v}$ that is even under each of its symmetry operations: $\sigma_{x},\sigma_{y},\sigma_{d1},\sigma_{d2},R_{4}$, and $R_{2}$. The constant contours of the 1DR wave functions nearest to the boundary are parallel to it, due to the Dirichlet boundary conditions.  |  |  |  |  |  |  |  $\sigma_{h}$, |  $\sigma_{d1}$,
---|---|---|---|---|---|---|---|---
Type | Symmetry | $\psi^{(\pm)}_{n,m}(x,y)$ | $n$ | $E$ | $R_{2}$ | $2R_{4}$ | $\sigma_{v}$ | $\sigma_{d2}$
$A_{1}$ | $x^{2}+y^{2}$ | $\Psi_{n,n},\Psi_{n,n+2p}^{(+)}$ | o | 1 | +1 | +1 | +1 | +1
$A_{2}$ | $xy(x^{2}-y^{2})$ | $\Psi_{n,n+2p}^{(-)}$ | e | 1 | +1 | +1 | -1 | -1
$B_{1}$ | $x^{2}-y^{2}$ | $\Psi^{(-)}_{n,n+2p}$ | o | 1 | +1 | -1 | +1 | -1
$B_{2}$ | $xy$ | $\Psi_{n,n},\Psi_{n,n+2p}^{(+)}$ | e | 1 | +1 | -1 | -1 | +1
$A_{1}$* | point nodes | $\Psi^{(\theta,\pm)}_{n,n+2p+1}$ | e,o | 2 | -2 | 0 | 0 | 0
$A_{1}$* | nodal squares† | $\Psi^{(\theta,\pm)}_{n,n+2p+1}$ | e,o | 2 | -2 | 0 | 0 | 0
Table 1: Square box representation types, symmetries, allowed 1DRs
$\Psi_{n,n}(x,y)$ and $\Psi_{n,n+2p}^{(\pm)}(x,y)$ [Eq. (7)] for odd or even
$n\geq 1$, 2DRs $\Psi^{(\theta,\pm)}_{n,n+2p+1}(x,y)$ [Eq. (6)] of the square
box, and operations of the $C_{4v}$ point group. $\sigma_{h}$, $\sigma_{v}$,
$\sigma_{d1}$, and $\sigma_{d2}$ are the mirror planes along the horizonal and
vertical axes and the two diagonals, $R_{n}$ represents rotations by $2\pi/n$
about the centroid, and $E$ is the trace of the identity matrix for the
appropriate group dimension. For the 2DR wave functions, the listed elements
are the traces of the rank-2 matrices that describe the operations Tinkham .
*Note that for the 2DR wave functions with no symmetry, their sets of common
point nodes and square nodes all have $A_{1}$ symmetry. †Under special
conditions. See Figs. 3 and 4 and the text. Figure 4: Presentations of
selected 2DRs with dots and squares. (a) $\Psi_{3,6}(x,y),\Psi_{6,3}(x,y)$.
(b) $\Psi_{3,12}(x,y),\Psi_{12,3}(x,y)$. (c)
$\Psi_{9,12}(x,y),\Psi_{12,9}(x,y)$.

For $m=n+2p$, there are again two classes. For $n$ odd, $\Psi_{n,n+2p}(x,y)$
is even under reflections about $\sigma_{h}$, $\sigma_{v}$ and under $R_{2}$,
but shows no symmetry under $R_{4}$ and about $\sigma_{d1}$, $\sigma_{d2}$.
For $n$ even, $\Psi_{n,n+2p}(x,y)$ is odd under reflections about
$\sigma_{x}$, $\sigma_{y}$ and under $R_{2}$, but shows no symmetry under
$R_{4}$ and about $\sigma_{d1}$, $\sigma_{d2}$. However, we note that for both
$n$ odd or even, $R_{4}\Psi_{n,n+2p}(x,y)=\Psi_{n+2p,n}(x,y)$. This implies
that there are two orthonormal members of each subgroup,

$\displaystyle\Psi^{\pm}_{n,n+2p}(x,y)$ $\displaystyle=$
$\displaystyle[\Psi_{n,n+2p}(x,y)\pm\Psi_{n+2p,n}(x,y)]/\sqrt{2},$ (10)

which are doubly degenerate. These wave functions are also displayed in Fig.
3.

We note that $\Psi_{n,n+2p}^{+}(x,y)$ for $n$ odd is invariant under all
$C_{4v}$ operations, as seen in Fig. 3 for $\Psi_{1,3}^{+}(x,y)$. Hence, such
wave functions are in symmetry group $A_{1}$, and are listed as such in Table
I. For $n$ even, $\Psi^{+}_{n,n+2p}(x,y)$ is even under reflections about
$\sigma_{d1}$, $\sigma_{d2}$ and under $R_{2}$, but is odd under reflections
about $\sigma_{x}$, $\sigma_{y}$ and under the 2 $R_{4}$, so it is in symmetry
group $B_{2}$. For $n$ odd, $\Psi^{-}_{n,n+2p}(x,y)$ is even under reflections
about $\sigma_{h}$, $\sigma_{v}$ and $R_{2}$, but odd under reflections about
$\sigma_{d1}$, $\sigma_{d2}$ and under $R_{4}$, so it is in symmetry group
$B_{1}$. Finally, for $n$ even, $\Psi^{-}_{n,n+2p}(x,y)$ is odd under
reflections about all four mirror planes and under $R_{4}$, but is even under
$R_{2}$, so it is in symmetry group $A_{2}$. These classifications are all
listed in Table I.

In addition to the common nodal point structure of each infinitely degenerate
2DR wave function, some higher index 2DR wave functions have mutual nodal
squares, some examples of which are shown in Fig. 4. For such common nodal
squares to appear in the common nodal set of 2DRs, the lower quantum number
$n\geq 3$ must be odd. The lowest energy case is therefore the (3,6) case
pictured in Fig. 4(a). In Fig. 4(b), the (3,12) case is shown. In addition, a
more complicated nodal pattern is obtained for the (9,12) case pictured in
Fig. 4(c). In this case, both numbers factor into 3 times an odd or an even
number, and this factorization allows for the square common nodal structure,
each of which encloses a finite set of common nodal dots. Obviously as the
lower odd number increases, the common nodal patterns become increasingly
complicated. But it is noteworthy that in every set of common nodal dots
and/or squares, that set is invariant under all operations of $C_{4v}$, and
hence obeys the $A_{1}$ symmetry table.

##  III The square microstrip antenna

For a thin square microstrip antenna of the same geometry as for the square
box, with its origin at the lower left corner, but satisfying the EM wave
equation for $A_{z}(x,y,t)$, the normalized transverse magnetic wave functions
at a fixed time with the appropriate Neumann boundary conditions,

$\displaystyle\frac{\partial\Psi_{n,m}(x,y)}{\partial x}\Bigl{|}_{x=0,a}$
$\displaystyle=$ $\displaystyle\frac{\partial\Psi_{n,m}(x,y)}{\partial
y}\Bigl{|}_{y=0,a}=0,$

have the form

$\displaystyle\Psi_{n,m}(x,y)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lc}\frac{2}{a}\cos(n\pi x/a)\cos(m\pi
y/a),&m,n\geq 1\\\ \frac{\sqrt{2}}{a}\cos(n\pi x/a),&m=0,n\geq 1\\\
\frac{\sqrt{2}}{a}\cos(m\pi y/a),&n=0,m\geq 1\end{array}\right.$ (15)

The enhanced emission frequencies $f_{n,m}$ from the square thin MSA are

$\displaystyle f_{n,m}$ $\displaystyle=$ $\displaystyle
c_{0}\frac{\sqrt{n^{2}+m^{2}}}{2an_{\rm r}},$ (17)

where $c_{0}$ is the vacuum speed of light and $n_{\rm r}$ is the index of
refraction, which for Bi2212 IJJ-THz emitter devices that are on the order of
1 $\mu$m thick, is about 4.2. The case $n=m=0$ must be excluded, since light
must have a finite frequency. In this case, the color-coded lowest frequency
square MSA wave functions are shown in Fig. 5.

Figure 5: (color online) Color-coded plots of the square MSA $\Psi_{n,m}(x,y)$ given by Eq. (LABEL:MSAwavefunctions) for $0\leq n,m\leq 3$, which are the nominal representations of the lowest frequency wave functions of a square microstrip antenna with Neumann boundary conditions that are indicated by the red boundaries. $n$ and $m$ are respectively the column and row numbers. $\Psi_{0,0}=1/a$ shown in blue corresponds to frequency $f_{0,0}=0$, so it is excluded. The color code bar otherwise applies to each of these figures. |  |  |  |  |  |  |  $\sigma_{h}$, |  $\sigma_{d1}$,
---|---|---|---|---|---|---|---|---
Type | Symmetry | $\psi^{(\pm)}_{n,n^{\prime}}(x,y)$ | $n$ | $E$ | $R_{2}$ | $2R_{4}$ | $\sigma_{v}$ | $\sigma_{d2}$
$A_{1}$ | $x^{2}+y^{2}$ | $\Psi_{n,n},\Psi_{n,n+2p}^{(+)}$ | e | 1 | +1 | +1 | +1 | +1
$A_{2}$ | $xy(x^{2}-y^{2})$ | $\Psi_{n,n+2p}^{(-)}$ | o | 1 | +1 | +1 | -1 | -1
$B_{1}$ | $x^{2}-y^{2}$ | $\Psi^{(-)}_{n,n+2p}$ | e | 1 | +1 | -1 | +1 | -1
$B_{2}$ | $xy$ | $\Psi_{n,n},\Psi_{n,n+2p}^{(+)}$ | o | 1 | +1 | -1 | -1 | +1
$A_{1}$ | point nodes* | $\Psi^{(\theta,\pm)}_{n,n+2p+1}$ | e,o | 2 | -2 | 0 | 0 | 0
Table 2: Square MSA representation types, symmetries, allowed 1DRs
$\Psi_{n,n}(x,y)$ and
$\Psi_{n,n+2p}^{(\pm)}(x,y)=[\Psi_{n,n+2p}(x,y)\pm\Psi_{n+2p,n}(x,y)]/\sqrt{2}$
for odd or even $n\geq 1$, 2DRs $\Psi^{(\theta,\pm)}_{n,n+2p+1}(x,y)$, which
have the forms of Eq. (6), except that their components satisfy Eq. (9), and
operations of the $C_{4v}$ point group. For the 2DR wave functions, the listed
values are the traces of the rank-2 matrices that describe the operations
Tinkham . *Note that the 2DR wave functions have no symmetry, but their sets
of common fixed point nodes have $A_{1}$ symmetry. See text Klemm2017b .

As for the square box wave functions, the $n=m$ MSA wave functions are all
non-degenerate 1DRs of the $C_{4v}$ point group, and the $m=n+2p+1$ MSA wave
functions are infinitely degenerate 2DRs of $C_{4v}$, having the forms of Eqs.
(6) and (9), except that the wave function components are given by Eq.
(LABEL:MSAwavefunctions) instead of Eq. (4). This second point is illustrated
in Fig. 6. In addition, since the MSA wave functions satisfy
$\Psi_{n,n+2p}(x,y)=R_{4}\Psi_{n+2p,n}(x,y)$, exactly as for the square box,
these wave functions are likewise doubly degenerate, and satisfy Eq. (10),
although again with the wave function components given by Eq.
(LABEL:MSAwavefunctions). The symmetry table of the square MSA wave functions
is therefore shown in Table II. The only differences between Table II and
Table I for the square box is that the oddness or evenness of the quantum
number $n$ is precisely the opposite, and there are no nodal squares for the
square MSA 2DR wave functions. The matrices that describe the symmetry
operations upon the 2DR MSA wave functions are identical to those presented
for the square quantum box wave functions following Eq. (9).

We then redisplay those and additional MSA wave functions in the array shown
in Fig. 7. As for the square box, the diagonal $n=m$ square MSA wave functions
are all nondegenerate 1DRs of $C_{4v}$, and are displayed in color-coded
contour plots. The infinitely degenerate $m=n+2p+1$ cases are again displayed
as a set of mutually common nodal points, but in this case, as discussed in
more detail in the following, there are no mutual nodal lines at any higher
index number. Furthermore, the $m=n+2p$ cases are also doubly degenerate, with
the $\Psi^{\pm}_{n,n+2p}(x,y)$ given by Eq. (10) with the appropriate MSA wave
functions, and with the $\Psi^{+}_{n,n+2p}$ displayed above the diagonal, and
the $\Psi^{-}_{n,n+2p}$ displayed below the array diagonal, exactly as in Fig.
3 for the box.

Figure 6: (color online) (a) Color-coded plots of the square MSA 2DR
$\Psi^{-\theta,+}_{0,1}(x,y)$ for $\theta=30^{\circ},45^{\circ}$, and
$60^{\circ}$ from left to right. (b) Plot of
$\Psi_{0,1}(x,y)+R_{2}\Psi_{0,1}(x,y)$, which equals 0. Figure 7: (color
online) Shown are plots of the array of square MSA wave functions
$\Psi_{n,m}(x,y)$ with Neumann boundary conditions for $0\leq n,m\leq 4$. 1DR
wave functions of nondegenerate $\Psi_{n,n}$ (for $n\geq 1$) and doubly-
degenerate $\Psi_{n,n+2p}^{\pm}$ are shown in color-coded plots along the
array diagonal and in the appropriate positions above (below) that diagonal.
2DR Plots of the infinitely-degenerate $\Psi_{n,n+2p+1}$ are shown as arrays
of black mutual nodal points. The constant contours nearest to a boundary
intersect it normally due to the Neumann boundary conditions.

Fr square box 2DR wave functions with $n,m>0$ without any line nodes, the
number of point nodes is $N(n,m)=(n-1)^{2}+(m-1)^{2}$. Similarly, for $n,m\geq
0$, the number of square MSA point nodes in a 2DR is $N(n,m)=n^{2}+m^{2}$.
However, as argued in the following, no nodal lines or squares appear in the
square MSA 2DRs. But as for the square box, the set of common nodal points for
each 2DR is invariant under all operations of $C_{4v}$, and hence obeys the
$A_{1}$ symmetry table.

A nodal line could occur along the $x$ direction if $\cos(n\pi x/a)=0$ for
$0<x<a$ and along the $y$ direction if $\cos(m\pi y/a)=0$ for $0<y<a$. These
require $x/a=(2p+1)/(2n)$ and $y/a=(2q+1)/(2m)$ for integer $p$ and $q$. For
them to occur together and the pattern to be invariant under the operations of
$C_{4v}$, including $R_{4}$, which interchanges $x$ and $y$, we then require

$\displaystyle\frac{2p+1}{2n}$ $\displaystyle=$
$\displaystyle\frac{2q+1}{2m}.$ (18)

As noted from Table II, if $n=m$, $\Psi_{n,m}(x,y)$ is a non-degenerate 1DR,
and if $n$ and $m$ are either both odd or both even, then $\Psi_{n,m}(x,y)$ is
a doubly-degenerate combination of two 1DRs. But if either $n$ or $m$ is odd
and the other is even, then $\Psi_{n,m}(x,y)$ is a 2DR. Rewriting Eq. (11) as
$(2p+1)m=(2q+1)n$, it is easy to see this cannot be satisfied with either $n$
or $m$ odd and the other even.

##  IV The equilateral triangular box

Previous studies have focused on the 1DR wave functions of equilateral
triangular MSAs Cerkoney2017 . Here we calculate the quantum wave functions
and normalization constants for the equilateral triangle of side $a$ in an
infinite potential well. The Schrödinger equation admits even and odd wave
function solutions about any of the symmetry axes. Here we choose the
horizontal axis as the axis of symmetry from which the wave functions can be
generated. The equilateral triangular box wave functions even and odd about
that axis may be written as

$\displaystyle\Psi^{e}_{\ell,m,n}(x,y)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!A^{e}_{m,n}\biggl{\\{}\sin\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)\ell\right]$ (19)
$\displaystyle\times\cos\left[\frac{2\pi(m-n)y}{3a}\right]$
$\displaystyle+\sin\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)m\right]\cos\left[\frac{2\pi(n-\ell)y}{3a}\right]$
$\displaystyle+\sin\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)n\right]\cos\left[\frac{2\pi(\ell-m)y}{3a}\right]\biggr{\\}}\>\>\>\>\>\>\>\>$

and

$\displaystyle\Psi^{o}_{\ell,m,n}(x,y)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!A^{o}_{m,n}\biggl{\\{}\sin\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)\ell\right]$ (20)
$\displaystyle\times\sin\left[\frac{2\pi(m-n)y}{3a}\right]$
$\displaystyle+\sin\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)m\right]\sin\left[\frac{2\pi(n-\ell)y}{3a}\right]$
$\displaystyle+\sin\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)n\right]\sin\left[\frac{2\pi(\ell-m)y}{3a}\right]\biggr{\\}}.\>\>\>\>\>\>\>\>$

Since each of the three terms for the even and odd wave functions must satisfy
the Schrödinger equation
$-\frac{\hbar^{2}}{2M}{\bm{\nabla}}^{2}\Psi=E\Psi=\frac{\hbar^{2}k^{2}}{2M}\Psi=0$,
this is equivalent to the EM wave equation,
${\bm{\nabla}}^{2}\Psi+k^{2}\Psi=0$. Since for each wave function form, each
term must separately satisfy that wave equation, we find

$\displaystyle(n-m)(\ell+n+m)$ $\displaystyle=$ $\displaystyle 0,$ (21)

as for the equilateral MSA Cerkoney2017 . As for the antenna, the $n=m$ cases
can be shown to not produce any additional wave functions, so we assume
$\ell=-n-m$. However, in this case, both the odd and even wave functions
vanish on the entire equilateral triangular boundary.

The energies for a quantum particle of mass $M$ in an equilateral triangular
quantum box are then found to be

$\displaystyle E_{n,m}$ $\displaystyle=$
$\displaystyle\frac{(4\pi\hbar)^{2}}{2M(3a)^{2}}(m^{2}+n^{2}+mn),$ (22)

which is $\propto k^{2}$ for the equilateral triangular MSA. The corresponding
normalization coefficients are obtained by integrating
$|\Psi^{o,e}_{\ell,m,n}|^{2}$ with $\ell=-n-m$ over the area of the
equilateral triangle, and dividing by that area. We find

$\displaystyle A_{m,n}^{e}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{4}{3^{3/4}a},&\quad m>n\neq
0,n>m\neq 0\\\ \frac{2\sqrt{2}}{3^{3/4}a},&\quad m=n\neq 0\hskip
10.0pt,\end{array}\right.$ (25) $\displaystyle A_{m,n}^{o}$ $\displaystyle=$
$\displaystyle\frac{4}{3^{3/4}a},\quad m>n\neq 0,n>m\neq 0.$ (26)

These equilateral triangular box normalization constants are remarkably
similar to those obtained for the equilateral triangular MSA Cerkoney2017 , as
shown in Section IV.

Figure 8: (color online) Array of color coded plots of the lowest energy, normalized equilateral triangular box wave functions with $n,m=1,2,3,4$ from Eqs. (19), (20), and (26). The array diagonal consists of 1DR wave functions that are invariant under all $C_{3v}$ operations. The figures above and below the diagonal are respectively even and odd about the horizontal axis. The color code bar applies to each of these figures. Type | Symmetry | $\psi^{(e,o)}_{n,m}(x,y)$ | $E$ | $2R_{3}$ | $3\sigma_{v}$
---|---|---|---|---|---
$A_{1}$ | $x^{2}+y^{2}$ | $\Psi^{e}_{n,n+3p}(x,y)$ | 1 | +1 | +1
$A_{2}$ | $y(3x^{2}-y^{2})$ | $\Psi^{o}_{n,n+3p}(x,y)$ | 1 | +1 | -1
$A_{1}*$ | nodal points |  $\Psi^{(o,e)}_{n,m\neq n+3p}$, | 2 | -1 | 0
|  |  $n,m$ not both even |  |  |
$A_{1}*$ | nodal points |  $\Psi^{(o,e)}_{n,m\neq n+3p}$, $n,m$ both even | 2 | -1 | 0
| and triangles |  |  |  |
Table 3: Representation types, symmetries, allowed 1DRs
$\Psi^{e,o}_{n,n+3p}(x,y)$ and 2DRs $\Psi^{(o,e)}_{n,m\neq n+3p}(x,y)$ of the
equilateral triangular box, and operations of the $C_{3v}$ point group. For
1DR wave functions, there are 3 mirror planes $\sigma_{v}$ that bisect each
angle, two rotations $R_{3}$ by $\pm 2\pi/3$ about the centroid, and $E$ is
the trace of the identity matrix for the appropriate group dimension Tinkham .
The 2DR wave functions have only one $\sigma_{v}$. *common nodal structure.
See Figs. (9) - (11) and the text.

To illustrate examples of the equilateral triangular box wave functions that
are even or odd about only one vertex, we show pictorially that for 2DR
equilateral triangular box wave functions that

$\displaystyle\\!\\!|\Psi^{(e,o)}_{2,3}(x,y)\rangle\\!+\\!R_{3}|\Psi_{2,3}^{(e,o)}(x,y)\rangle\\!+\\!R_{3}^{2}|\Psi^{(e,o)}_{2,3}(x,y)\rangle$
$\displaystyle\\!=\\!$ $\displaystyle 0,\>\>\>\>$ (27)

where we have used the Dirac ket notation.

Figure 9: (color online) Even (a) and odd (b) examples of 2DR equilateral
triangular box wave functions pictured in Fig. 8 each satisfy the picture
equation $|\Psi\rangle+R_{3}|\Psi\rangle+R_{3}^{2}|\Psi\rangle=0$. In these
examples, $(n,m)=(2,3)$.

These equations show that only two of the wave functions even or odd about
only one vertex are linearly independent, demonstrating that these wave
functions are 2DR wave functions. Thus, we could choose as the general basis
$|\Psi^{(e,o)}_{n,m}(x,y)\rangle$ and $R_{3}|\Psi^{(e,o)}_{n,m}(x,y)\rangle$,
where $m\neq n+3p$.

But since these wave functions are not orthonormal, we set

$\displaystyle|\Psi_{n,m}^{(e,o,1)}(x,y)\rangle$ $\displaystyle\\!=\\!$
$\displaystyle
A|\Psi^{(e,o)}_{n,m}(x,y)\rangle\\!+\\!BR_{3}|\Psi^{(e,o)}(x,y)\rangle,\>\>\>$
$\displaystyle|\Psi_{n,m}^{(e,o,2)}(x,y)\rangle$ $\displaystyle\\!=\\!$
$\displaystyle
C|\Psi^{(e,o)}_{n,m}(x,y)\rangle\\!+\\!DR_{3}|\Psi^{(e,o)}(x,y)\rangle,\>\>\>$
(28)

for constants $A,B,C$, and $D$, and force them to form an orthonormal set.

To do so, we first take the inner product of Eq. (27) with
$\langle\Psi^{(e,o)}_{n,m}(x,y)|R_{3}^{{\dagger}}$, and it is easily seen that

$\displaystyle\langle\Psi^{(e,o)}_{n,m}(x,y)|R_{3}^{{\dagger}}\Psi^{(e,o)}_{n,m}(x,y)\rangle$
$\displaystyle=$ $\displaystyle-\frac{1}{2}.$ (29)

Although complex coefficients are possible, especially for
$|A|>\frac{2}{\sqrt{3}}$, under the assumption that all coefficients are real,
it is then easy to show that the orthonormal set may be written as

$\displaystyle|\Psi_{n,m\neq n+3p}^{(e,o,1,\theta)}(x,y)\rangle$
$\displaystyle=$
$\displaystyle\frac{2}{\sqrt{3}}\cos\theta|\Psi^{(e,o)}_{n,m}(x,y)\rangle$
$\displaystyle+\Bigl{(}\frac{1}{\sqrt{3}}\cos\theta\pm\sin\theta\Bigr{)}R_{3}|\Psi^{(e,o)}_{n,m}(x,y)\rangle,$
$\displaystyle|\Psi_{n,m\neq n+3p}^{(e,o,2,\theta)}(x,y)\rangle$
$\displaystyle\\!=\\!$
$\displaystyle-\frac{2}{\sqrt{3}}\sin\theta|\Psi^{(e,o)}_{n,m}(x,y)\rangle$
$\displaystyle\\!-\Bigl{(}\frac{1}{\sqrt{3}}\sin\theta\mp\cos\theta\Bigr{)}R_{3}|\Psi^{(e,o)}_{n,m}(x,y)\rangle,$

where $0\leq\theta<2\pi$ is arbitrary. As for the square quantum box with
$m=n+2p+1$ , the equilateral triangular quantum box wave functions with $m\neq
n+3p$ are 2DRs and are also infinitely degenerate.

When acting on the Nambu form of these 2DR wave functions,

$\displaystyle|\Psi^{(e,o,\theta)}_{n,m\neq n+3p}(x,y)\rangle$
$\displaystyle=$
$\displaystyle\Biggl{(}\begin{array}[]{c}|\Psi^{(e,o,1,\theta)}_{n,m\neq
n+3p}(x,y)\rangle\\\ |\Psi^{(e,o,2,\theta)}_{n,m\neq
n+3p}(x,y)\rangle\end{array}\Biggr{)},$ (33)

the matrices $R_{3}$ and $R_{3}^{{\dagger}}$ are

$\displaystyle R_{3}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}{\bm{1}}\pm
i\frac{\sqrt{3}}{2}\sigma_{y},$ (34)

and its Hermitian conjugate, both traces of which are -1, as indicated in
Table III.

With regard to the mirror symmetry operations of a 2DR wave function about a
single vertex, the even wave functions satisfy
$\sigma^{(e)}_{v}|\Psi^{(e,\theta)}_{n,m}\rangle=|\Psi^{(e,\theta)}_{n,m}\rangle$,
$\sigma^{(e)}_{v}R_{3}|\Psi^{(e,\theta)}_{n,m}\rangle=R_{3}^{2}|\Psi^{(e,\theta)}_{n,m}\rangle$
and
$\sigma^{(e)}_{v}R_{3}^{2}|\Psi^{(e,\theta)}_{n,m}\rangle=R_{3}|\Psi^{(e,\theta)}_{n,m}\rangle$,
as evidenced from Fig. 9(a). Combining these equations with Eq. (27), it is
then straightforward to show that the mirror plane matrix $\sigma_{v}^{(e)}$
when acting on the Nambu form of Eq. (LABEL:Phi2DREBox) may be written for the
even functions as

$\displaystyle\sigma_{v}^{(e)}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{z}}{2}\bigl{[}\cos(2\theta)\mp\sqrt{3}\sin(2\theta)\bigr{]}$
(35)
$\displaystyle-\frac{\sigma_{x}}{2}\bigl{[}\sin(2\theta)\pm\sqrt{3}\cos(2\theta)\bigr{]},$

which is traceless, as indicated in Table III. On other hand, the 2DR wave
functions odd about one vertex satisfy
$\sigma^{(o)}_{v}|\Psi^{(o,\theta)}_{n,m}\rangle=-|\Psi^{(o,\theta)}_{n,m}\rangle$,
$\sigma^{(o)}_{v}R_{3}|\Psi^{(o,\theta)}_{n,m}\rangle=-R_{3}^{2}|\Psi^{(o,\theta)}_{n,m}\rangle$
and
$\sigma^{(o)}_{v}R_{3}^{2}|\Psi^{(o,\theta)}_{n,m}\rangle=-R_{3}|\Psi^{(o,\theta)}_{n,m}\rangle$,
as sketched in Fig. 9(b). Again, combining these equations with Eq. (27), when
acting upon the odd Nambu form of Eq. (LABEL:Phi2DREBox),
$\sigma_{v}^{(o)}=-\sigma_{v}^{(e)}$, which is given by Eq. (35), so that both
traces of $\sigma_{v}^{(e)}$ and $\sigma_{v}^{(o)}$ vanish, as indicated in
Table III.

As for the square box, we then redraw the 2DR equilateral triangular box wave
functions in terms of their common set of nodes. This results in the array
pictured in Fig. 10. We note that $\Psi_{1,4}^{e}(x,y)$ and
$\Psi_{1,4}^{o}(x,y)$ pictured in the top right and bottom left array
positions are 1DRs, as are all four of the $\Psi_{n,n}^{e}(x,y)$ along the
array diagonal. In addition, $\Psi_{2,4}^{e}(x,y)$ and $\Psi_{2,4}^{o}(x,y)$
are both 2DRs that contain an identical set of nodal points plus a single
equilateral triangular nodal figure in their center.

Figure 10: (color online) Plots of the same equilateral triangular box wave
functions pictured in Fig. 8, but displaying the 2DR wave functions in terms
of their loci of common nodes. Figure 11: Shown are some examples of higher
index equilateral triangular box 2DR wave functions represented by both common
points and equilateral triangles. (a) $\Psi^{(e,o)}_{2,4}$. (b)
$\Psi^{(e,o)}_{2,6}$. (c) $\Psi^{(e,o)}_{4,6}$. (d) $\Psi^{(e,o)}_{4,8}$.

We note that both with and without the equilateral triangular nodal lines
inside the box, the loci of points and lines in each 2DR is invariant under
all of the operations of $C_{3v}$. Therefore, we classify those nodal loci as
having symmetry $A_{1}$ in Table III. We note that the common internal
equilateral triangular nodal figures only arise for $(n,m\neq n+3p)$ both
even.

##  V The equilateral triangular microstrip antenna

Type | Symmetry | $|\Psi^{(e,o)}_{n,m}(x,y)\rangle$ | $E$ | $2R_{3}$ | $3\sigma_{v}$
---|---|---|---|---|---
$A_{1}$ | $x^{2}+y^{2}$ | $|\Psi^{e}_{n,n+3p}(x,y)\rangle$ | 1 | +1 | +1
$A_{2}$ | $y(3x^{2}-y^{2})$ | $|\Psi^{o}_{n,n+3p}(x,y)\rangle$ | 1 | +1 | -1
$A_{1}*$ | fixed point nodes | $|\Psi^{(o,e,\theta)}_{n,m\neq n+3p}\rangle$ | 2 | -1 | 0
Table 4: Representation types, symmetries, allowed 1DRs
$\Psi^{e,o}_{n,n+3p}(x,y)$ and 2DRs $\Psi^{(o,e)}_{n,m\neq n+3p}(x,y)$ of the
thin equilateral triangular MSA, and operations of the $C_{3v}$ point group.
For the 1DR wave functions, there are 3 mirror planes $\sigma_{v}$ that bisect
each angle, two rotations $R_{3}$ by $\pm 2\pi/3$ about the centroid, and $E$
is the trace of the identity matrix for the appropriate group dimension. For
the 2DR wave functions, there are the same rotations, but only one
$\sigma_{v}$ Tinkham . *common nodal points. See Fig. 13 and text. Figure 12:
(color online) Array of color-coded plots of the lowest frequency equilateral
triangular microstrip antenna wave functions, generated from Eqs.
(LABEL:etmsaeven), (LABEL:etmsaodd), and (46), with $n,m=0,1,2,3$. The upper
left solid blue figure has $f_{0,0}=0$, so it can be excluded. The diagonal of
the array represents $\Psi^{e}_{n,n}(x,y)$, which for $n>0$ are 1DRs. The
$n\neq m$ even and odd wave functions lie respectively above and below the
array diagonal. The red boundaries indicate the Neumann conditions. The color
code bar applies to each of these figures except the excluded (0,0) case.

Although the wave functions for the thin equilateral triangular MSA were given
previously Cerkoney2017 , those authors only plotted the 1DR wave functions,
and calculated the angular distributions of the output power from those
resonant cavity modes and from the uniform Josephson current source at those
mode frequencies. Here we are primarily interested in contrasting the
pictorial representation of the 1DR and 2DR wave functions. We have

$\displaystyle\Psi^{e}_{\ell,m,n}(x,y)$ $\displaystyle\\!=\\!$ $\displaystyle
A^{e}_{m,n}\\!\biggl{\\{}\cos\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)\ell\right]$
$\displaystyle\times\cos\left[\frac{2\pi(m-n)y}{3a}\right]\>\>\>\>\>\>$
$\displaystyle+\cos\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)m\right]\cos\left[\frac{2\pi(n-\ell)y}{3a}\right]\>\>\>\>\>\>\>$
$\displaystyle+\cos\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)n\right]\cos\left[\frac{2\pi(\ell-m)y}{3a}\right]\biggr{\\}}\>\>\>\>\>\>$

and

$\displaystyle\Psi^{o}_{\ell,m,n}(x,y)$ $\displaystyle\\!=\\!$ $\displaystyle
A^{o}_{m,n}\biggl{\\{}\cos\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)\ell\right]$
$\displaystyle\times\sin\left[\frac{2\pi(m-n)y}{3a}\right]$
$\displaystyle+\cos\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)m\right]\sin\left[\frac{2\pi(n-\ell)y}{3a}\right]$
$\displaystyle+\cos\left[\left(\frac{2\pi
x}{\sqrt{3}a}+\frac{2\pi}{3}\right)n\right]\sin\left[\frac{2\pi(\ell-m)y}{3a}\right]\biggr{\\}}.\>\>\>\>\>\>\>$

Each of the three terms for the even and odd wave functions must satisfy the
EM wave equation ${\bm{\nabla}}^{2}\Psi+(k^{\prime})^{2}\Psi=0$, as for the
square MSA. These forms can be shown to also obey the Neumann boundary
conditions. As for the equilateral triangular box, we again have

$\displaystyle(n-m)(\ell+n+m)$ $\displaystyle=$ $\displaystyle 0,$ (38)

and the same arguments for $n=m$ in Section IV lead to the conclusion
$\ell=-n-m$Cerkoney2017 .

The emission frequencies $f_{n,m}$ from an equilateral triangular MSA with
index of refraction $n_{\rm r}$ are then found to be

$\displaystyle f_{n,m}$ $\displaystyle=$ $\displaystyle\frac{2c_{0}}{3an_{\rm
r}}\sqrt{m^{2}+n^{2}+mn}.$ (39)

The corresponding normalization coefficients are obtained by integrating
$|\Psi^{o,e}_{\ell,m,n}|^{2}$ over the area of the equilateral triangle, and
dividing by that area. As was found previously Cerkoney2017 ,

$\displaystyle A_{m,n}^{e}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{4}{3^{3/4}a},&\quad m,n\geq
1,m\neq n\\\ \frac{2\sqrt{2}}{3^{3/4}a},&\quad m>n=0,n>m=0,{\rm or}\hskip
2.0ptm=n\hskip 10.0pt,\end{array}\right.$ (42) $\displaystyle A_{m,n}^{o}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{4}{3^{3/4}a},&\quad m,n\geq
1,m\neq n,\\\ \frac{2\sqrt{2}}{3^{3/4}a},&\quad m>n=0,{\rm or}\hskip
2.0ptn>m=0,\\\ 0,&\quad m=n.\end{array}\right.$ (46)

As shown in the previous section, these normalization constants are remarkably
similar to those obtained for the equilateral triangular box.

Figure 13: (color online) (a) $|\Psi^{e}_{3,4}(x,y)\rangle$ and odd (b)
$|\Psi^{o}_{3,4}(x,y)\rangle$ examples of 2DR equilateral triangular MSA wave
functions not pictured in Fig. 12 that each satisfy the picture equation
$|\Psi\rangle+R_{3}|\Psi\rangle+R_{3}^{2}|\Psi\rangle=0$

As for the 2DR wave functions for the equilateral triangular box that satisfy
Eq. (27), the 2DR wave functions for the thin equilateral triangular MSA
exhibit the same symmetries. For example, in Fig. (13), we show pictorially
that

$\displaystyle\\!\\!|\Psi^{(e,o)}_{3,4}(x,y)\rangle+R_{3}|\Psi^{(e,o)}_{3,4}(x,y)\rangle+R_{3}^{2}|\Psi^{(e,o)}_{3,4}(x,y)\rangle$
$\displaystyle\\!=\\!$ $\displaystyle 0,\>\>\>\>\>\>\>$ (47)

the only difference being the wave functions are not the box wave functions
with Dirichlet boundary conditions given by Eqs. (12)-(16), but are instead
given by Eqs. (24)-(28) for the MSA, which satisfy the Neumann boundary
conditions with the normal derivative vanishing on each of the triangle’s
sides. Therefore, the 2DR wave functions can be constructed exactly by analogy
with Eqs. (18)-(21), also leading the analogous Nambu representation and the
rank 2 matrices representing the identical symmetry operations $R_{3}$ and
$\sigma_{v}^{(e,o)}$ given by Eqs. (22) and (23). These thin equilateral MSA
wave functions are also infinitely degenerate, as they also contain the
arbitrary analogous mixing angle $\theta$.

Figure 14: (color online) Plots of the same equilateral triangular MSA wave
functions pictured in Fig. 12, but displaying the 2DR wave functions in terms
of their loci of common nodes.

Thus, of the equilateral triangular MSA wave functions pictured, those in
color are 1DRs, and those along the array diagonal and in the upper right
figure have $A_{1}$ symmetry, and the figure in the lower left has $A_{2}$
symmetry. The rest of the figures are 2DRs, and the pattern of nodal points
has $A_{1}$ symmetry that is invariant under all operations of point group
$C_{3v}$. The matrices representing the 2$R_{3}$ and single $\sigma_{v}$
operations of the $C_{3v}$ point group acting upon the 2DR wavefunctions are
identical to those described for the equilateral triangular box, except that
the spatial parts of the wave functions are given by Eqs. (LABEL:etmsaeven)
and (LABEL:etmsaodd).

We remark that the only difference between Tables III and IV for the
equilateral triangular box and the thin equilateral triangular MSA is that
some (with $m$ and $n$ both even) of the 2DR box wave functions contain both
common nodal points and internal equilateral triangles (as well as on the
boundary), but the 2DR MSA wave functions only contain common nodal points. In
both cases, the loci of the sets of nodal points and/or triangles are
invariant under all operations of point group $C_{3v}$.

##  VI The cylindrical box

Figure 15: (color online) Color coded plots of the 16 lowest energy
cylindrical box wave functions, all oriented with $\theta=0$ and ranked in
order from the top left to the bottom right array positions, listed as
$(m,p)$: Top row: (0,1), (1,1), (2,1), (0,2). Second row: (3,1), (1,2), (4,1),
(2,2). Third row: (0,3), (5,1), (3,2), (6,1). Fourth row: (1,3), (4,2), (7,1),
(2,3). Each figure has a qualitatively similar but numerically distinct color
code bar. The color code bar shown is for the (2,2) mode.

For the quantum particle in a cylindrical box of radius $a$, the Schrödinger
equation is written in polar coordinates. $V(\rho)=0$ for $0\leq\rho<a$, and
$V(\rho)=\infty$ for $\rho\geq a$. Using separation of variables and assuming
$\Psi(\rho,\phi+2\pi)=\Psi(\rho,\phi)$, one obtains the Bessel equation with
solutions of the first kind. Since $\Psi(\rho,\phi)$ must be finite inside the
cylindrical box, we only have the integer Bessel functions of the first kind,
$J_{m}(k_{m}\rho)$ multiplied by $\sin(m\phi)$ or $\cos(m\phi)$. Therefore a
general state may be written

$\displaystyle\Psi_{m}(\rho,\phi)=[B_{m}\cos(m\phi)+C_{m}\sin(m\phi)]J_{m}(k_{m}\rho).\>\>$
(48)

For the disk box of radius $a$, we require $\Psi_{m}(a,\phi)=0$, or

$\displaystyle J_{m}(k_{m}a)$ $\displaystyle=$ $\displaystyle 0.$ (49)

Since there are many possible zeroes of $J_{m}(x)$, we index them with
$k_{m,p}$ values. Thus, we set

$\displaystyle J_{m}(k_{m,p}a)$ $\displaystyle\\!=\\!$ $\displaystyle 0,$ (50)
$\displaystyle\Psi_{m,p}(\rho,\phi)$ $\displaystyle\\!=\\!$
$\displaystyle[B_{m,p}\cos(m\phi)+C_{m,p}\sin(m\phi)]J_{m}(k_{m,p}\rho).\>\>\>\>\>\>\>\>$
(51)

It is immediately obvious that the cases $m=0$ and $m\geq 1$ are qualitatively
different. For $m=0$, the wave functions $\Psi_{0,p}(\rho)$ are 1DRs
independent of $\phi$. For $m\geq 1$, the wave functions are all 2DRs. Since
$\cos(m\phi)$ and $\sin(m\phi)$ are orthogonal when integrated over $\phi$
from 0 to $2\pi$, we could write either $B_{m,p}=A_{m,p}\cos(m\theta)$ and
$C_{m,p}=A_{m,p}\sin(m\theta)$ or $B_{m,p}=-A_{m,p}\sin(m\theta)$ and
$C_{m,p}=A_{m,p}\cos(m\theta)$. Thus, we have two infinitely degenerate wave
functions, which in Nambu form can be written as

$\displaystyle\Psi^{\theta}_{m,p}(\phi,\rho)$ $\displaystyle=$
$\displaystyle\Biggl{(}\begin{array}[]{c}\Psi^{(\theta,1)}_{m,p}(\phi,\rho)\\\
\Psi^{(\theta,2)}_{m,p}(\phi,\rho)\end{array}\Biggr{)}$ (54) $\displaystyle=$
$\displaystyle
A_{m,p}J_{m}(k_{m,p}\rho)\Biggl{(}\begin{array}[]{c}\cos[m(\phi-\theta)]\\\
\sin[m(\phi-\theta)]\end{array}\Biggr{)},\>\>$ (57)

where $\theta$ satisfying $0\leq\theta<2\pi$ is a 2DR wave function mixing
angle as for the 2DR wave functions of the square and equilateral triangular
boxes, and the $A_{m,p}$ are found by normalization of both
$|\Psi^{(1)}_{m,p}(\rho,\phi)|^{2}$ and
$|\Psi^{(2)}_{m,p}(\rho,\phi)|^{2}$over the cross-sectional area of the
cylinder,

$\displaystyle A_{m,p}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lc}\frac{1}{a\sqrt{\pi\int_{0}^{1}xdxJ_{m}^{2}(\chi_{m,p}x)}},&m\geq
1\\\
\frac{1}{a\sqrt{2\pi\int_{0}^{1}xdxJ_{0}^{2}(\chi_{0,p}x)}},&m=0\end{array}\right.,$
(60) $\displaystyle\chi_{m,p}$ $\displaystyle=$ $\displaystyle k_{m,p}a.$ (61)

Figure 16: (color online) Color-coded plots of $\Psi_{2,2}(\rho,\phi)$ with
$\theta=0^{\circ}$ (left), and $\theta=53^{\circ}$ (right).

The energy of the $(m,p)$ mode for a spinless quantum particle of mass $M$ in
the cylindrical box is given by

$\displaystyle E_{m,p}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}\chi_{m,p}^{2}}{2Ma^{2}}.$ (62)

Figure 17: Representing the cylindrical box wave functions in Fig. 15 to
emphasize the differences between the non-degenerate $\Psi_{0,p}(\rho,\phi)$
and the infinitely degenerate $\Psi_{m,p}(\rho,\phi)$ for $m\geq 1$.

A list of $\chi_{m,p}$ values for $0\leq m\leq 8$ and $1\leq p\leq 4$ is given
in Table V.

$m$ | $\chi_{m,1}$ | $\chi_{m,2}$ | $\chi_{m,3}$ | $\chi_{m,4}$
---|---|---|---|---
0 | 2.4048 | 5.52007 | 8.65372 | 11.79153
1 | 3.8317 | 7.01558 | 10.1734 | 13.32369
2 | 5.13562 | 8.41724 | 11.61984 | 14.795981
3 | 6.38016 | 9.76102 | 13.01520 | 16.22346
4 | 7.58834 | 10.64709 | 14.3725 |
5 | 8.77148 | 12.3386 | 15.70017 |
6 | 9.936109 | 13.58929 | 17.0038 |
7 | 11.08637 | 14.821268 |  |
8 | 12.22509 | 16.03777 |  |
Table 5: Table of $\chi_{m,p}=k_{m,p}a$ for the cylindrical box of radius $a$.
Missing table entries correspond to higher energy states than pictured in Fig.
15.

In Fig. 15, we have presented an array of the 16 lowest energy wave functions
for a quantum particle in a cylindrical box. However, from Eq. (48), it is
evident that the angle $\theta$ is arbitrary. Since $0\leq\theta<2\pi$, it can
take on an infinite number of values, and hence cylindrical box eigenstates
with $m\neq 0$ are infinitely degenerate. In Fig. 16, this arbitrary
degeneracy is illustrated by comparing $\Psi_{2,2}(\rho,\phi)$ with $\theta=0$
and with its orientation with $\theta=53^{\circ}$. Hence, those cylindrical
box wave functions with straight line nodes passing through their centroids
are infinitely degenerate. Other than the circular line nodes at fixed $\rho$,
the straight line nodes that can be rotated only have the single node at their
common origin, the centroid. This is illustrated in Fig. 17.

In order to construct the symmetry table, we first note that for $m\geq 1$,
Eq. (57) contains two components, $\Psi^{(\theta,1)}_{m,p}(\phi,\rho)$ and
$\Psi^{(\theta,2)}_{m,p}(\phi,\rho)$, which form the orthonormal components of
a 2DR wave function. When the two rotation matrices $R_{\pm m\varphi}$ for
rotations by $\pm m\varphi$ about the $z$-axis normal to the centroid act on
this Nambu form for the 2DR wave function, they are easily found to be $R_{\pm
m\varphi}={\bm{1}}\cos(m\varphi)\pm i\sigma_{y}\sin(m\varphi)$, the traces of
which are $2\cos(m\varphi)$. This rotation matrix changes $\theta$ to
$\theta\pm\varphi$ in the Nambu representation. Since the bottom of the box is
not a symmetry plane, there are no reflection planes.

|  |  | circular |  |
---|---|---|---|---|---
Type | Symmetry | $m$ | nodes | E | $R_{\pm m\varphi}$
$A_{1}$ | $x^{2}+y^{2}$ | 0 | $p$ | 1 | 1
$A_{1}$* | centroid node | $\geq 1$ | $p$ | 2 | $2\cos(m\varphi)$
Table 6: Representation types, symmetries, and operations of the $C_{\infty
v}$ point group for the cylindrical box. The number of circular nodes depends
upon $p$. $R_{\pm m\varphi}$ is a rotation about the centroid by the angle
$\pm m\varphi$ and $E$ is the trace of the identity matrix for the appropriate
group dimension Tinkham . See Fig. 17 and text. *common nodal pattern. Figure
18: (color online) Lowest frequency wave functions for the disk MSA Klemm2010b
. $f_{m,p}$ values increase from the top left to the bottom right. Top row,
left to right: (1,1), (2,1), (0,1), (3,1). Second row: (4,1), (1,2), (5,1),
(2,2). Third row: (0,2), (6,1), (3,2), (1,3). Fourth row:
(7,1),(4,2),(8,1),(2,3). The red boundaries indicate the Neumann condition.
The color code bars are qualitatively similar but numerically distinct for
each of these figures. The color code shown is for the (2,1) mode.

##  VII The disk microstrip antenna

The thin disk microstrip antenna has been studied previously Klemm2010b . Here
we include it for two reasons: to compare the wave functions forms with those
of the cylindrical box, and to use the degeneracy of the low-energy wave
functions to correctly identify the experimentally measured resonant cavity
mode emitted from a disk Bi2212 IJJ-THz emitter, which will be described in
Section XIII.

For the disk MSA, the wave functions also have the form of Eq. (48), but the
boundary condition is different:

$\displaystyle\frac{dJ_{m}(k_{m}\rho)}{d\rho}\Bigr{|}_{\rho=a}$
$\displaystyle=$ $\displaystyle 0.$ (63)

As for the cylindrical box wave functions, there are an infinite number of
such wave functions, which have the same forms as in Eq. (25), but the
$\chi_{m,p}$ are different than those for the cylindrical box. The amplitudes
$A_{m,p}$ are also given by Eq. (61), but with the different $\chi_{m,p}$
values appropriate for the disk MSA. The emission frequencies $f_{m,p}$ from
the 1DR cavity modes of the disk MSA are given by Klemm2010b

$\displaystyle f_{m,p}$ $\displaystyle=$
$\displaystyle\frac{c_{0}\chi_{m,p}}{2\pi an_{\rm r}},$ (64)

where the lowest group of $\chi_{m,p}$ values are listed in Table VII.

$m$ | $\chi_{m,1}$ | $\chi_{m,2}$ | $\chi_{m,3}$ | $\chi_{m,4}$
---|---|---|---|---
0 | 3.8317 | 7.0156 | 10.1735 | 13.3237
1 | 1.8412 | 5.3314 | 8.5363 | 11.7060
2 | 3.0542 | 6.7061 | 9.9695 |
3 | 4.2012 | 8.0152 | 11.3459 |
4 | 5.3176 | 9.2824 |  |
5 | 6.4156 | 10.5199 |  |
6 | 7.5013 | 11.7349 |  |
7 | 8.5778 |  |  |
8 | 9.6474 |  |  |
9 | 10.7114 |  |  |
10 | 11.7709 |  |  |
Table 7: Table of $\chi_{m,p}=k_{m,p}a$ for the disk microstrip antenna of
radius $a$. Missing table entries correspond to higher energy states than
pictured in Fig. 18.  Figure 19: (color online) Comparison of
$\Psi_{2,2}(\rho,\phi)$ of the disk MSA with $\theta=0^{\circ}$ (left) and
$\theta=31^{\circ}$ (right). Figure 20: (color online) Plots of the same disk
MSA wave functions as in Fig. 18, but with the straight nodal lines passing
though the centroid represented by a dot at the centroid for all of the 2DR
wave functions.

In conclusion, for the cylindrical box and disk MSA, there are only two types
of wave functions: Non-degenerate wave functions with $m=0$ that have no nodal
lines passing through the centroid, and a much larger class of infinitely
degenerate wave functions with $m\geq 1$ that have one or more nodal lines
passing through the centroid. When there is no reason to prefer one nodal line
direction, the degeneracy is random, with an infinite number of possible
directions. The rotation matrices $R_{\pm m\varphi}$ for the 2DR MSA wave
functions are identical to Eq. (57) for the 2D cylindrical box, with the MSA
$k_{m,p}$ values, the lowest energy values of which are given by Table VII.
The only difference in symmetry Tables VI and VIII for the cylindrical box and
the thin disk MSA is in the number of circular nodes for their 2DR wave
functions, which is one more for the box due to the boundary condition. Thus,
we conclude that for the thin MSAs, the only modes that can build up a cavity
resonance are those of the 1DR wave functions with $m=0$.

|  |  | circular |  |
---|---|---|---|---|---
Type | Symmetry | $m$ | nodes | E | $R_{\pm m\varphi}$
$A_{1}$ | $x^{2}+y^{2}$ | 0 | $p$ | 1 | 1
$A_{1}$* | centroid node | $\geq 1$ | $p-1$ | 2 | $2\cos(m\varphi)$
Table 8: Representation types, symmetries, and operations of the $C_{\infty
v}$ point group for the disk MSA wavefunctions $\Psi_{m,p}(\phi,\rho)$. The
number of circular nodes depends upon $p$. The $R_{\pm m\varphi}$ are
rotations about the $z$ axis normal to the centroid by $\pm m\varphi$ and $E$
is the trace of the identity matrix for the appropriate group dimension
Tinkham . See Fig. 20 and text. *common nodal pattern. Figure 21: Frequency
dependence of the emission from a stand-alone Bi2212 disk mesa Kashiwagi2015 .
Reprinted with permission of T. Kashiwagi, K. Sakamoto, H. Kubo, Y. Shibano,
T. Enomoto, T. Kitamura, K. Asanuma, T. Yasui, C. Watanabe, K. Nakade, Y.
Saiwai, T. Katsuragawa, M. Tsujimoto, R. Yoshizaki, T. Yamamoto, H. Minami, R.
A. Klemm, and K. Kadowaki, A high-$T_{c}$ intrinsic Josephson junction emitter
tunable from 0.5 to 2.4 terahertz, Appl. Phys. Lett. 107, 082601 (2015). ©2015
AIP Publishing LLC. Figure 22: Frequency dependence of the emission from a
square stand-alone Bi2212 mesa Sun2018 . Reprinted with permission of H. Sun,
R. Wieland, Z. Xu, Z. Qi, Y. Lv, Y. Huang, H. Zhang, X. Zhou, J. Li, Y. Wang,
F. Rudau, J. S. Hampp, D. Koelle, S. Ishida, H. Eisaki, Y. Yoshida, B. Jin, V.
P. Koshelets, R. Kleiner, H. Wang, and P. Wu, Compact high-$T_{c}$
superconducting terahertz emitter operating up to 86 K, Phys. Rev. Appl. 10,
024041 (2018). ©2018 American Physical Society.

##  VIII Comparison with experiments

Since the original discovery of coherent THz emission from the IJJs in Bi2212
Ozyuzer2007 , a variety of experimental groups in many countries have been
working on the topic, trying to understand its properties and to increase the
output power. There have so far been six review articles on the subject
Kashiwagi2012 ; Welp2013 ; Kakeya2016 ; Kashiwagi2017 ; Kleiner2019 ;
Delfanazari2020 . In the early work, the first type of Bi2212 mesas were
formed by subjecting a cleaved single crystal of Bi2212 to an Ar beam with a
mask, that cut into the unmasked region of the crystal, leaving a standing
mesa with the remainder of the Bi2212 crystal as the substrate. A second type
was a groove mesa, obtained by simply cutting a groove into the top of a
cleaved Bi2212 crystal, which was first done for groove rectangular, square,
and disk mesas Tsujimoto2010 , and subsequently for a groove equilateral
triangular mesa Delfanazari2013 . In those experiments, the emission
frequencies for 3 groove disk mesas, the rectangular, the square, and 3 groove
equilateral triangular mesas were all consistent with their respectively
lowest-frequency TM(1,1) and TM(0,1) modes Tsujimoto2010 ; Delfanazari2013 .
For the rectangular mesa, the TM(0,1) mode with a nodal line bisecting the
mesa length is non-degenerate, and as expected, was the first shape to be
shown to build up a cavity resonance Ozyuzer2007 . However, as discussed in
Sections II, V, and VII, those lowest frequency TM(0,1) square and equilateral
triangular modes and the lowest frequency TM(1,1) disk modes are all
infinitely degenerate, and if the groove mesas were sufficiently accurately
constructed for those symmetries to be relevant, shouldn’t build up a cavity
resonance at those frequencies Klemm2010a ; Klemm2010b . The facts that the
lowest frequency, infinitely-degenerate cavity modes were observed in those
experiments were most likely due to the breaking of the square, equilateral
triangular, or disk symmetry due to some phenomenon that was not understood at
the time of those early experiments.

The problem turned out to be that the introduction of a dc $V$ and current $I$
across the stack of IJJs in Bi2212 leads to severe Joule heating effects,
resulting in hot spots, spatial regions in which $T>T_{c}$ Wang2009 ; Wang2010
; Minami2014 ; Kashiwagi2017b . For a rectangular Bi2212 mesa, these hot spots
were observed by laser scanning Wang2009 ; Wang2010 , SiC photoluminescence
Minami2014 , and thermoreflectance microscopy Kashiwagi2017b , and when a hot
spot develops away from the center of a square, equilateral triangular, or
disk mesa, it breaks the symmetry, and allows for the emission of photons at
the lowest-frequency from the infinitely-degenerate respective modes. But it
was suggested that removing the mesa from its superconducting substrate and
coating the top and bottom with a perfect electric conductor such as Au, the
output power could be enhanced Klemm2010a ; Klemm2010b . After doubly cleaving
a Bi2212 sample from a single crystal, the top and bottom surfaces are first
covered with about 50-100 nm of Ag, followed by about 50-100 nm of Au. These
mesas with thin Au layers on their top and bottom surfaces are presently known
as either as “stand-alone” mesas Kashiwagi2015b , or as “GBG” for “gold-
Bi2212-gold” mesas Sun2018 . Since Au is a superior thermal conductor, as long
as the stand-alone mesas are not much thicker than 1-2 $\mu$m, it is usually
possible to avoid most of the heating problems, including the development of
hot spots. An efficient procedure to manufacture the stand-alone Bi2212 mesas
was published Kashiwagi2015b . In such mesas, the amount of Joule heating was
greatly reduced, and it became possible to investigate experimentally the
effects of mesa symmetry upon the cavity resonances observed.

In Fig. 21, the frequency dependence of the emission from a Bi2212 stand-alone
disk mesa is shown. Unlike the emission data from three groove disk mesas
studied earlier Tsujimoto2010 , the stand-alone disk mesa did not build up a
cavity resonance at the infinitely degenerate TM(1,1) disk MSA mode. Instead,
a strong cavity resonance appeared at 1.0 THz, in-between the predicted
resonance frequencies of the TM(0,1) and TM(2,1) modes. However, since the
stand-alone Bi2212 disk mesa was sandwiched in-between two sapphire
substrates, the substrates might cause a slight shift in the cavity resonance
frequency. Although we are not aware of calculations for a MSA sandwiched
between two substrates, when a MSA is sitting atop a sapphire substrate, there
is a slight downward shift in the emission frequency Balanis . In addition,
since the infinitely degenerate TM(1,1) mode was not excited, it seems
reasonable to assume that the infinitely degenerate TM(2,1) mode would also
not be excited. For those two reasons, we assign the strong emission at 1 THz
to the non-degenerate TM(0,1) disk MSA mode. From the data, it appears that
the downward frequency shift due to the two sapphire substrates is
approximately 3% for emission at 1.0 THz.

More recently, the emissions from a stand-alone square Bi2212 mesa with
$a=200\mu$m sandwiched between sapphire substrates was studied Sun2018 , and
the frequency dependence of the emission from that mesa is shown in Fig. 22.
From Eq. (11), it is possible to analyze the emission spectrum in terms of the
possible mode frequencies. We note that the authors originally misidentified
the cavity resonance mode indices, but corrected them in an erratum Sun2018 .
A table of mode indices, the degeneracies, and calculated frequencies without
and with a 3% substrate effect is given in Table IX. We note that the lowest
two frequencies are at or below the low-temperature Josephson plasma frequency
$f_{p}\approx 0.250$ THz Singley2004 , and are screened out by the Josephson
plasma.

$(n,m)$ | $g$ |  $f_{n,m}$ (THz) |  $f^{\prime}_{n,m}$ (THz)
---|---|---|---
(1,0) | $\infty$ | 0.179* | 0.174*
(1,1) | 1 | 0.253* | 0.245*
(2,0) | 2 | 0.357 | 0.346
(1,2) | $\infty$ | 0.399 | 0.387
(2,2) | 1 | 0.505 | 0.490
(3,0) | $\infty$ | 0.536 | 0.520
(3,1) | 2 | 0.564 | 0.547
(3,2) | $\infty$ | 0.644 | 0.625
(4,0) | 2 | 0.714 | 0.693
(4,1) | $\infty$ | 0.736 | 0.714
(3,3) | 1 | 0.758 | 0.735
(4,2) | 2 | 0.799 | 0.775
(4,3) | $\infty$ | 0.892 | 0.865
(5,0) | $\infty$ | 0.892 | 0.865
(5,1) | 2 | 0.910 | 0.883
(5,2) | $\infty$ | 0.962 | 0.933
(4,4) | 1 | 1.010 | 0.980
(5,3) | 2 | 1.04 | 1.01
Table 9: Predicted cavity mode frequencies $f_{n,m}$ from Eq. (10), the
estimated $f^{\prime}_{n,m}=0.97f_{n,m}$ due to the substrate in THz, the
degeneracy $g$ of the $(n,m)$ mode for a square stand-alone Bi2212 mesa of
side $200\mu$m and $n_{\rm r}=4.2$. *At or below the Josephson plasma
frequency $f_{p}\approx 0.250$ THz Singley2004 .

It is not clear that one should assume the substrate reduction factor to be
the same percentage for each frequency measured. But the resonances near to
1.0 THz are likely to have nearly the same shift in the two experiments. Hence
a strong case can be made that the resonances are mostly associated with the
non-degenerate $(n,n)$ modes. It is certainly true that the doubly degenerate
$(n,n+2p)$ modes have orthogonal 1DR symmetries, so on some time scale, it
would be difficult for the system to stick with one symmetry and to ignore the
other one. That is, if the system oscillates between the two symmetries on a
time scale inverse to the common mode frequency, there would be no cavity
resonance. The data are consistent with this notion Sun2018 .

##  IX Summary and conclusions

We have studied the wave functions of high-symmetry two-dimensional quantum
boxes and electromagnetic microstrip antennas (or cavities). The symmetries
studied are those of a square, an equilateral triangle, and a disk. Each of
these symmetries has one- and two-dimensional representations of its wave
functions. The two-dimensional representations are infinitely degenerate
Tinkham . In addition, for square boxes and microstrip antennas, there are
doubly-degenerate cases, the wave functions of which can be written in terms
of orthogonal one-dimensional representations.

Although the two-dimensional quantum box might have some approximate
experimental relevance to quantum wells, the main interest from the
experimental side is for thin microstrip antennas. This is particularly of
importance for the coherent THz emission from the intrinsic Josephson
junctions in the layered high-temperature superconductor, Bi2Sr2CaCu2O8+δ, or
Bi2212. In the early days of the coherent THz emission from Bi2212, there were
severe heating effects that interfered with accurate comparisons of theory
with experiment. Now that thermally-managed stand-alone Bi2212 (or Au-
Bi2212-Au) mesas are the primary devices under study, the effect of the
degeneracies of the modes is important to consider. The comparison of the
experimental output from stand-alone Bi2212 disk and square mesas provide
experimental evidence that the non-degenerate modes are the ones that can be
excited in order to increase the output power. This suggests the construction
of stand-alone mesas that have only or predominantly non-degenerate modes.

Figure 23: (a) Sketch of an annular stand-alone mesa with a single slit. (b)
Sketch of an array of 84 stand-alone mesas. Seven stand-alone disk mesas are
fixed in a hexagonal close packed array, and are equally cut circularly with
an atomic beam into smaller disks and annulli. Then, with straight line slits,
the original disks are each cut into 12 mesas, and the entire array of 84
mesas could be close enough to one another to emit coherently. Delfanazari2020

The simplest example is that of a rectangle in which the ratio of length to
width is not that of two integers, stand-alone mesas of which showed
excitations at many frequencies Kashiwagi2018 . A singly-slitted annulus has
been suggested as another possibility Delfanazari2020 , and our independent
analysis has shown that the modes odd and even about the slit are not
degenerate Shouk2021 . But another possibility is a disk that is cut into 12
pieces of two different types: The disk is first cut into a smaller disk and
an annulus by a He or Ar beam, and those two objects are cut with three
straight cuts at angles 60∘ apart, dividing the smaller disk and annulus into
six equivalent pie-shaped wedge mesas and six equivalent hexaslitted annuli.
These would all fit closely together, allowing for the possibility of coherent
emission from a much larger number of intrinsic Josephson junctions,
increasing the output power well above 1 mW. Our preliminary studies showed
that all of the modes are nondegenerate.

Such a device has been proposed recently Delfanazari2020 , and is redrawn in
Fig. 23. In order to maximize the probability of matching the resonant
frequencies of the inner pie-shaped wedge mesas with the outer hexaslitted
annular mesas, the ratio of $\rho_{i}/\rho_{o}$ is varied, and the resonant
frequencies can be calculated to find if at least two frequencies from both
shapes will match in the 1-2 THz range for which the output power of compact
continuous wave coherent sources is generally less than the 1 mW necessary for
practical applications Delfanazari2020 .

Since the only equilateral triangular microstrip antennas of Bi2212 were made
prior to the construction of stand-alone mesas Delfanazari2013 ;
Delfanazari2013b , they most likely exhibited hot spots, breaking the symmetry
and allowing the infinitely-degenerate TM(0,1) mode to be observed. We
therefore encourage experiments on stand-alone equilateral triangular Bi2212
mesas to compare with our predictions, as was done with stand-alone disk and
square mesas Kashiwagi2015 ; Sun2018 .

##  X Acknowledgments

The authors acknowledge discussions with Kazuo Kadowaki and Allan H.
MacDonald. R. A. K. was supported in part by the U. S. Air Force Office of
Scientific Research (AFOSR) LRIR #18RQCOR100, and the AFRL/SFFP Summer Faculty
Fellowship Program provided by AFRL/RQ at WPAFB.

J. R. R., P. Y. C., and A. B. contributed equally to this work. R. A. K.
supervised the work and wrote the paper.

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