# Multimode parity-time and loss-compensation symmetries in coupled waveguides
with loss and gain

Anton V. Hlushchenko  State Key Laboratory of Integrated Optoelectronics,
College of Electronic Science and Engineering, International Center of Future
Science, Jilin University, 2699 Qianjin Street, Changchun, 130012, China
National Science Center ‘Kharkiv Institute of Physics and Technology’ of
National Academy of Sciences of Ukraine, 1, Akademicheskaya Street, Kharkiv
61108, Ukraine  Vitalii I. Shcherbinin  National Science Center ‘Kharkiv
Institute of Physics and Technology’ of National Academy of Sciences of
Ukraine, 1, Akademicheskaya Street, Kharkiv 61108, Ukraine  Denis V. Novitsky
B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus,
68 Nezavisimosti Avenue, Minsk 220072, Belarus  Vladimir R. Tuz
<EMAIL_ADDRESS>State Key Laboratory of Integrated Optoelectronics, College of
Electronic Science and Engineering, International Center of Future Science,
Jilin University, 2699 Qianjin Street, Changchun, 130012, China

###### Abstract

Loss compensation via inserting gain is of fundamental importance in different
branches of photonics, nanoplasmonics, and metamaterial science. This effect
has found an impressive implementation in the parity-time symmetric
($\mathcal{PT}$-symmetric) structures possessing balanced distribution of loss
and gain. In this work, we generalize this phenomenon to the asymmetric
systems demonstrating loss compensation in the coupled multi-mode loss-gain
dielectric waveguides of different radii. We show that similar to the
$\mathcal{PT}$-symmetric coupled single-mode waveguides of identical radii,
the asymmetric systems support the exceptional points called here the loss
compensation (LC) thresholds where the frequency spectrum undergoes a
transition from complex to real values. Moreover, the LC-symmetry thresholds
can be obtained for dissimilar modes excited in the waveguides providing an
additional degree of freedom to control the system response. In particular,
changing loss and gain of asymmetric coupled waveguides, we observe loss
compensation for TM and TE modes as well as for the hybrid HE and EH modes.

##  I Introduction

The intriguing properties of loss-gain distributions are the subject of the
fast-growing field of non-Hermitian photonics. The most notable and studied
class of non-Hermitian systems is the $\mathcal{PT}$-symmetric systems which
possess the perfectly balanced spatial distribution of gain and loss [1, 2,
3]. $\mathcal{PT}$ symmetry guarantees the reality of non-Hermitian
Hamiltonian spectra [4] that have been readily implemented and observed in the
optical domain [5, 6]. Subsequently, $\mathcal{PT}$ symmetry was transferred
to electronic circuits [7], acoustics [8], and time-varying (Floquet) systems
[9].

The balance between loss and gain helps to solve the problem of loss
compensation which is of paramount importance for the efficiency of plasmonic
and metamaterial-based devices [10, 11, 12]. The loss compensation effect
proved to be connected to another remarkable feature of non-Hermitian systems
– the possibility of specific degeneracies called exceptional points (EPs)
which were observed in photonics [13, 14] as well as in acoustics [15]. In
contrast to the degeneracies of Hermitian systems (the so-called diabolic
points) with coalescing eigenvalues, the EPs imply coalescence of both
eigenvalues and eigenfunctions. In the $\mathcal{PT}$-symmetric systems, the
EPs mark the points of phase transitions between the $\mathcal{PT}$-symmetric
phase and the phase with spontaneously broken $\mathcal{PT}$ symmetry. It is
important to emphasize, however, that the EPs are the general phenomenon
observed also in purely passive systems (such as whispering-gallery-mode
microresonators [16], ring cavities [17], and anisotropic waveguides [18]) and
even in the systems with radiative loss only [19].

The EPs have become a workhorse of many recent achievements such as enhanced
perturbation sensing [20, 21], novel lasing schemes [22, 23], enhanced Sagnac
effect for laser gyroscopes [24, 25], simultaneous coherent perfect absorption
and amplification [26, 27], asymmetric transmission [5, 28], etc. An
interesting feature of EPs is their topological nature which can be revealed
with their dynamical encircling in parameter space and can be used for mode
switching [29], mode transfer [30], and polarization conversion [31]. Another
exciting direction is the observation and utilization of the higher-order EPs,
where more than two eigenmodes coalesce. Such EPs can be realized either in
the systems containing three or more resonant elements [21, 32, 33] or by
hybridizing several usual (second-order) EPs [34, 35]. It should be noted that
the most considerations of non-Hermitian effects are limited to the single-
mode case with the use of the coupled-mode theory (CMT) [36]. The multi-mode
platforms provide much richer opportunities in controlling optical response
and dynamics as evidenced by the examples of dispersion engineering and mode
conversion in the waveguide and microresonator systems [37, 38, 39].

For the symmetric system of coupled dielectric waveguides, the full loss
compensation is possible only for the perfect balance between gain and loss
corresponding to the non-violated $\mathcal{PT}$-symmetry [6]. This poses
rather tough and difficult-to-achieve conditions for experimental realization
of the related effects [13]. On the contrary, for the asymmetric system, the
full loss compensation can be reached even for the unbalanced gain and loss.
By analogy with the $\mathcal{PT}$-symmetry, this generalized situation can be
called the loss-compensation (LC) symmetry [40, 41]. In this paper, we propose
further generalization demonstrating LC symmetry for the coupled dissimilar
waveguides under excitation of the modes with the same or different azimuthal
indices. In dielectric waveguides, all modes except TE and TM are hybrid,
i.e., they have axial components of both electric and magnetic fields.
Therefore, to analyze the system, we apply the multi-mode approach [42, 41]
which allows us to obtain exact solutions of the eigenvalue problem for all
possible classes of modes supported by the asymmetric guiding structure. We
show that such an asymmetric structure supports specific EPs called the LC-
symmetry thresholds and reached for different loss-gain ratios depending on
the modes used. Thus, the mode composition of the non-Hermitian system is an
additional degree of freedom useful for tuning the parameters necessary to
obtain the constant-intensity modes [43, 44] and other applications.

##  II Multimode analytical approach

For our analysis, we use the multi-mode analytical approach [42], which was
previously successfully applied to coupled systems with loss and gain [41].
Following Ref. [41], we consider a pair of coupled dielectric cylinders of the
radii $R_{1}$ and $R_{2}$ placed in an ambient medium. Figure 1 shows the
cross-sections of the dielectric circular waveguides with the permittivities
$\varepsilon_{1}$ and $\varepsilon_{2}$, respectively (Regions I and II). The
polar coordinate systems associated with the waveguides are denoted as
($r_{1},\phi_{1}$) and ($r_{2},\phi_{2}$). The waveguides are placed in the
infinite uniform medium (Region III) with the real permittivity
$\varepsilon_{3}\in\Re$. The permeability $\mu=\mu_{0}$ is assumed to be the
same in all regions. Without loss of generality, we suppose that the first
waveguide contains gain material and the second one contains lossy medium
(Im$(\varepsilon_{1})<0$ and Im$(\varepsilon_{2})>0$), whereas
Re$(\varepsilon_{1})>\varepsilon_{3}$ and
Re$(\varepsilon_{2})>\varepsilon_{3}$.

Figure 1:  Geometry of unequally sized dielectric waveguides immersed in an
infinite medium.

In order to solve the eigenvalue problem for this system, we write the
electromagnetic fields in every region using the corresponding polar
coordinates as shown in Fig. 1. For example, the $E_{z}$ and $H_{z}$
components of electromagnetic fields in the Region I are written in terms of
the local coordinates $(r_{1},\phi_{1})$ as

$\begin{split}E_{z}^{1}&=\sum_{n=-N}^{N}{A_{n}^{1}J_{n}(k_{p,1}r_{1})e^{in\phi_{1}}},\\\
H_{z}^{1}&=\sum_{n=-N}^{N}{B_{n}^{1}J_{n}(k_{p,1}r_{1})e^{in\phi_{1}}},\end{split}$
(1)

whereas in the Region II the local coordinates $(r_{2},\phi_{2})$ are used as

$\begin{split}E_{z}^{2}&=\sum_{n=-N}^{N}{A_{n}^{2}J_{n}(k_{p,2}r_{2})e^{in\phi_{2}}},\\\
H_{z}^{2}&=\sum_{n=-N}^{N}{B_{n}^{2}J_{n}(k_{p,2}r_{2})e^{in\phi_{2}}},\end{split}$
(2)

and, finally, in the Region III we have the contributions from both local
coordinate systems,

$\begin{split}E_{z}^{3}=&\sum_{n=-N}^{N}{C_{n}^{1}H_{n}^{(1,2)}(k_{p,3}r_{1})e^{in\phi_{1}}}\\\
&+\sum_{n=-N}^{N}{C_{n}^{2}H^{(1,2)}_{n}(k_{p,3}r_{2})e^{in\phi_{2}}},\\\
H_{z}^{3}=&\sum_{n=-N}^{N}{D_{n}^{1}H^{(1,2)}_{n}(k_{p,3}r_{1})e^{in\phi_{1}}}\\\
&+\sum_{n=-N}^{N}{D_{n}^{2}H^{(1,2)}_{n}(k_{p,3}r_{2})e^{in\phi_{2}}}.\end{split}$
(3)

Here
$\\{A_{n}^{1},A_{n}^{2},B_{n}^{1},B_{n}^{2},C_{n}^{1},C_{n}^{2},D_{n}^{1},D_{n}^{2}\\}$
are the unknown amplitudes of azimuthal harmonics,
$k_{p,i}^{2}=k^{2}_{i}-k_{z}^{2}$, $k_{i}=k_{0}\varepsilon_{r_{i}}$
$(i=1,2,3)$, $\varepsilon_{r_{i}}=\varepsilon_{i}/\varepsilon_{0}$ is the
relative permittivity, $k_{0}^{2}=\omega^{2}\varepsilon_{0}\mu_{0}$,
$J_{n}(\cdot)$ is the Bessel function, $H_{n}^{(1)}(\cdot)$ and
$H_{n}^{(2)}(\cdot)$ are the Hankel functions of the first and second kind,
respectively, and the field factor of the form $\exp{[-i(\omega t-k_{z}z)]}$
is assumed and omitted. The choice of the Hankel function is governed by the
boundary conditions for the fields in the Region III, as follows:
$E_{z}^{3}\rightarrow 0$ and $H_{z}^{3}\rightarrow 0$ for
$r_{1},r_{2}\rightarrow\infty$.

Since the derivation of other field components is cumbersome, it is relegated
to the Appendix. We only note that the Maxwell equations and Eqs. (1)-(3)
allow us to obtain the expressions for the electromagnetic field components
$E_{\phi}$ and $H_{\phi}$ in every Region. The unknown coefficients
$\\{A_{n}^{1},A_{n}^{2},B_{n}^{1},B_{n}^{2},C_{n}^{1},C_{n}^{2},D_{n}^{1},D_{n}^{2}\\}$
and $k_{z}$ can be obtained from the dispersion relation derived using the
continuity conditions for the tangential fields at the boundary surfaces
$r_{1}=R_{1}$ and $r_{2}=R_{2}$ (see Appendix).

##  III TM and TE modes

###  III.1 Single waveguide modes

Our aim is to demonstrate the possibility of full loss compensation for the
coupled dielectric waveguides with gain and loss supporting different modes
with an arbitrary value of azimuthal index. The dispersion relations of
coupled waveguides will tend to the dispersion relations of independent
waveguides with increasing distance between them. Therefore, we are interested
in the crossing points of the dispersion curves of two independent cylinders.
The dispersion curves for the lowest-order modes of a single waveguide are
shown in Fig. 2. Changing the radius or permittivity of another cylinder, one
can observe the shift of its dispersion curves with respect to the curves of
the first one. Thus, for any pair of modes, we can find their crossing points
at a required frequency $f$ and longitudinal wavenumber $k_{z}/k_{0}$. We will
show further that these crossing points are convenient for the realization of
full loss compensation at the certain values of gain-loss parameter and
distance between the cylinders.

Figure 2:  Lowest-order modes of the circular dielectric waveguide with
$\epsilon_{r}=12$ and $R=10$ $\mu$m

Let us consider, for the moment, the family of transverse electric TE0m and
transverse magnetic TM0m modes of the circular cylindrical waveguide. As
representatives of these modes, we choose the set {TM01, TM02, TE01, TE02}.
For example, for the TM01 (TE01) mode, we can choose an arbitrary solution of
the dispersion relation ($f$; $k_{z}/k_{0}$). Then, we increase the cylinder
radius to obtain same solution for the TM02 (TE02) mode. The initial and
increased radii of the waveguide can serve as a first approximation to find
the conditions of loss compensation for the coupled cylinders with gain and
loss. For instance, according to Fig. 2, the values $f=5.6$ THz and
$k_{z}/k_{0}=1.6286$ correspond to the TM01 (TE01) mode of the waveguide with
$\epsilon_{r1}=12$ and $R_{1}=10$ $\mu$m and to the TM02 (TE01) mode of the
waveguide with $\epsilon_{r2}=12$ and $R_{2}=18.93$ $\mu$m ($R_{2}=16.305$
$\mu$m).

###  III.2 Symmetric coupled waveguides: $\mathcal{PT}$ symmetry

Figure 3:  The real and imaginary parts of the eigenvalues $k_{z}/k_{0}$ for
the TM modes as a function of the distance $d$ between the coupled waveguides.
LC- and $\mathcal{PT}$-symmetry thresholds of TM-type modes are shown with
arrows.

Figure 4:  The real and imaginary parts of the eigenvalues $k_{z}/k_{0}$ for
the TE modes as a function of the distance $d$ between the coupled waveguides.
LC- and $\mathcal{PT}$-symmetry thresholds of TM-type modes are shown with
arrows.

We start with the case of symmetric system consisting of the coupled
waveguides with the same radii and permittivities, i.e., $R_{1}=R_{2}$ and
$\epsilon_{r1}=\epsilon_{r2}$. In the case of balanced loss and gain, when the
loss tangents have the same absolute value,
$\tan{\delta_{1}}=-\tan{\delta_{2}}$, this system is a
$\mathcal{PT}$-symmetric one. $\mathcal{PT}$-symmetric systems can exist in
two states: a $\mathcal{PT}$-symmetric state with the loss exactly compensated
by gain, and the broken-$\mathcal{PT}$-symmetry state with the violated
compensation. In our case, the transition between these states can be realized
by changing the distance between the cylinders. The point of transition where
the $\mathcal{PT}$ symmetry gets broken is the EP, where the modes of the
system are degenerate. For example, in Fig. 4, the dashed and dash-dotted
lines correspond to the TM01 and TM02 modes of the coupled waveguides with
balanced gain and loss (the parameters are $R_{1}=R_{2}=10$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=-\tan{\delta_{2}}=5\cdot
10^{-4}$ for TM01 and $R_{1}=R_{2}=18.93$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$,
$\tan{\delta_{1}}=-\tan{\delta_{2}}=4.05\cdot 10^{-4}$ for TM02), whereas in
Fig. 4, the dashed and dash-dotted lines correspond to the TE01 and TE02 modes
of the system with another set of parameters ($R_{1}=R_{2}=7.48$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=-\tan{\delta_{2}}=5\cdot
10^{-4}$ for TE01 and $R_{1}=R_{2}=16.305$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$,
$\tan{\delta_{1}}=-\tan{\delta_{2}}=4.09\cdot 10^{-4}$ for TE02,
respectively). The EPs are denoted as the $\mathcal{PT}$-symmetry thresholds,
since below these points (for distances smaller than the threshold one) the
eigenvalues $k_{z}/k_{0}$ become purely real and the system falls into the
$\mathcal{PT}$-symmetric state. This state is preserved to the very connection
of the cylinders at $d=0$. It is important to note that due to the symmetry of
the system, the full loss compensation via the $\mathcal{PT}$-symmetric state
formation is observed only for the identical modes of both waveguides.

###  III.3 Asymmetric coupled waveguides: LC symmetry

Although $\mathcal{PT}$-symmetric systems allow perfect loss compensation, it
is often problematic to reach the ideal symmetry and the ideal loss-gain
balance in realistic situations. Here, we show that loss compensation can be
obtained in asymmetric systems with unequal coupled waveguides. Such situation
can be described as LC symmetry [40], which has much in common with
$\mathcal{PT}$ symmetry (e.g., the existence of EPs), but at the same time,
the requirements for system symmetry are strongly relaxed. Moreover, LC
symmetry is reached for a pair of different modes as will be illustrated
further.

Let us take the loss tangent of the first cylinder equal to
$\tan{\delta_{1}}=5\cdot 10^{-4}$ corresponding to the value for silicon [45].
Changing the gain tangent $\tan{\delta_{2}}$ and the distance between the
cylinders $d$, we can obtain the full loss compensation at the target
frequency $f=5.6$ THz and wavenumber $k_{z}/k_{0}=1.6286$. For the TE01 and
TE02 modes, the EP which we call the LC-symmetry threshold is reached at the
system parameters as follows: $R_{1}=7.48$ $\mu$m, $R_{2}=16.305$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $d=26.67$ $\mu$m,
$\tan{\delta_{2}}=-4.05\cdot 10^{-4}$) (see Fig. 5, green and black lines).
For the modes TM01 and TM02, the EP is reached at $d=24.37$ $\mu$m and
$\tan{\delta_{2}}=-4.05\cdot 10^{-4}$ (see Fig. 5, blue and red lines).
Similarly to the $\mathcal{PT}$-symmetry threshold, the LC-symmetry threshold
corresponds to the purely real eigenvalues $k_{z}/k_{0}$. This approach can be
utilized to obtain the EPs of full loss compensations for other pairs of modes
as well.

Figure 5:  EPs corresponding to the LC symmetry for the pairs of modes
TM01-TM02 and TE01-TE02. Figure 6:  (a), (c) Electric $|E_{z}|$ and Magnetic
$|H_{z}|$ field components corresponding to the broken LC-symmetry case for
TM01 and TE01 loss modes at distance $d=35~{}\mu$m. (b), (d) Electric
$|E_{z}|$ and Magnetic $|H_{z}|$ field components corresponding to the LC-
symmetry thresholds of TM01-TM02 and TE01-TE02 mode pairs at distances
$d=24.37~{}\mu$m and $d=26.67~{}\mu$m. (e), (f) Electric field $|E_{z}|$
corresponding to the asymmetry and $\mathcal{PT}$-symmetry cases of TM01-TM01
mode pairs at distances $d=35~{}\mu$m and $d=26.31~{}\mu$m.

In Figs. 4 and 4, the LC-symmetry thresholds are shown for the TM-type and TE-
type modes, respectively, and compared to the $\mathcal{PT}$-symmetry
thresholds. One can see that the LC-symmetry thresholds for the different
modes (for example, TM01 and TM02) lie between the $\mathcal{PT}$-symmetry
thresholds for the corresponding individual modes. The main difference with
the $\mathcal{PT}$-symmetric case is that the eigenvalues for the asymmetric
waveguides do not remain real below the LC-symmetry threshold. In fact, full
loss compensation is realized only in a single point. The position of this
point – LC-symmetry threshold – depends on the loss tangent and the frequency
at which the dispersion curves cross. An example of electric and magnetic
field distributions at the LC-symmetry threshold and dots of broken symmetry
for dissimilar waveguides are shown in Figs. 6(b) and 6(d) and Figs. 6(a) and
6(c) as compared to the distributions at the $\mathcal{PT}$-symmetry
thresholds and case of broken $\mathcal{PT}$-symmetry for the identical
waveguides in Fig. 6(f) and 6(e). The smaller the loss tangent and the nearer
the crossing to the cutoff frequency, the larger the distance between the
waveguides corresponding to the threshold [41].

Thus, the LC-symmetry threshold in the asymmetric system allows one to fully
compensate the loss for the unbalanced gain and loss of the individual
cylinders. This effect can be considered as a generalization of
$\mathcal{PT}$-symmetric loss compensation, since both the dispersion curves
and the behavior of electromagnetic fields show that the nature of loss
compensation phenomenon in asymmetric systems is similar to the $\mathcal{PT}$
symmetry in symmetric ones.

##  IV Hybrid modes: LC-symmetry

###  IV.1 Modes with the same azimuthal index

We have shown above how LC and $\mathcal{PT}$ symmetries can be observed by
coupling either TM or TE modes of the pair of cylindrical waveguides. However,
when one mode is TM one and another is TE one, we cannot obtain an EP due to
the weak coupling between the modes of different types (see Fig. 4, green and
pink lines). In order to get around this problem, in this section, we consider
the possibility of LC symmetry for the hybrid modes, which can be treated as a
linear superposition of the corresponding TE and TM modes. These modes having
both the electric and magnetic field components in the longitudinal direction
can be either of HE type (magnetic component dominates) or EH type (electric
component dominates).

Figure 7:  LC symmetry for the hybrid modes with the azimuthal index $1$.

Figure 8:  (a), (b), (c) Electric field $|E_{z}|$ corresponding to the LC-
symmetry thresholds of HE11-HE12, HE11-EH11 and EH11-HE12 mode pairs; (d),
(e), (f) The corresponding magnetic field $|H_{z}|$.

Figure 9:  LC-symmetry of hybrid modes with the azimuthal index $2$.

Figure 10:  (a), (b), (c) Electric field $|E_{z}|$ corresponding to the LC-
symmetry thresholds of HE22-HE21, EH21-HE21 and HE22-EH21 mode pairs; (d),
(e), (f) The corresponding magnetic field $|H_{z}|$.

We tune the dispersion curves for two cylinders to obtain the LC-symmetry
thresholds of hybrid modes for the same parameters ($f=5.6$ THz and
$k_{z}/k_{0}=1.6286$) as for the TM and TE modes. As previously, we start from
the modes of the individual waveguide and shift the dispersion curves by
changing the radius to get the required mode at the target frequency and
wavenumber. For example, we obtain the hybrid mode HE11 at $f=5.6$ THz and
$k_{z}/k_{0}=1.6286$ for the cylinder with $R=5.732$ $\mu$m (see Fig. 2,
dashed line).

Let us consider the class of hybrid modes with azimuthal index $1$ focusing in
particular on the modes HE11, EH11 and HE12 (see Fig. 2, black lines). We are
interested in the LC symmetry between the mode pairs HE11-EH11, HE11-HE12 and
EH11-HE12 with the first mode corresponding to the cylinder with loss (Region
I) and the second mode corresponding to the cylinder with gain (Region II). As
shown in Fig. 10, the LC-symmetry thresholds exist for all the mode pairs
mentioned above and for the system parameters as follows:

  * •

HE11-EH11 – {$R_{1}=5.732$ $\mu$m, $R_{2}=11.44$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=5\cdot 10^{-4}$,
$\tan{\delta_{2}}=-4.91\cdot 10^{-4}$};

  * •

HE11-HE12 – {$R_{1}=5.732$ $\mu$m, $R_{2}=14.487$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=5\cdot 10^{-4}$,
$\tan{\delta_{2}}=-4.33\cdot 10^{-4}$};

  * •

EH11-HE12 – {$R_{1}=11.44$ $\mu$m, $R_{2}=14.487$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=4.91\cdot 10^{-4}$,
$\tan{\delta_{2}}=-4.33\cdot 10^{-4}$}.

We see that the LC-symmetry thresholds are reached for different distances
between waveguides for every pair of modes: $d=20.89$ $\mu$m for HE11-EH11;
$d=27.28$ $\mu$m for HE11-HE12; and $d=16.18$ $\mu$m for EH11-HE12. The field
profiles shown in Fig. 10 prove that there are nonzero $E_{z}$ and $H_{z}$ in
both cylinders as expected for the hybrid modes (although contribution of
electric or magnetic field can strongly differ).

As to the hybrid modes with the azimuthal index $2$, we choose the mode pairs
as follows: HE22-EH21, HE22-HE21 and EH21-HE21. Now, unlike previous cases,
the radius of the waveguide with loss should be taken larger than the radius
of the cylinder with gain. For the parameters of the corresponding LC-symmetry
thresholds, shown in Figs. 10 and 10, we have:

  * •

HE22-EH21 – {$R_{1}=18.7$ $\mu$m, $R_{2}=15$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=5\cdot 10^{-4}$,
$\tan{\delta_{2}}=-5.95\cdot 10^{-4}$};

  * •

HE22-HE21 – {$R_{1}=18.7$ $\mu$m, $R_{2}=9.91$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{(\delta_{1})}=5\cdot 10^{-4}$,
$\tan{(\delta_{2})}=-4.65\cdot 10^{-4}$};

  * •

EH21-HE21 – {$R_{1}=15$ $\mu$m, $R_{2}=9.91$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=5\cdot 10^{-4}$,
$\tan{\delta_{2}}=-4\cdot 10^{-4}$}.

The distances between the waveguides corresponding to these thresholds are:
$d=16.64$ $\mu$m for HE22-EH21; $d=16.95$ $\mu$m for HE22-HE21; and $d=20.56$
$\mu$m for EH21-HE21. The results in Figs. 10-10 evidence that the full loss
compensation can be realized for the arbitrary hybrid modes with the same
azimuthal index. The structure of the modes influence only which of the
cylinders should have larger radius – the lossy or the gainy one.

###  IV.2 Modes with different azimuthal indices

In this subsection, we study the possibility of LC symmetry in the coupled
waveguides with loss and gain tuned to the modes with different azimuthal
indices. The fundamental difference to the cases considered above is the
necessity to take into account the larger number of azimuthal harmonics. For
the identical azimuthal indices from the previous subsection, one can obtain
the LC-symmetry threshold with just one harmonic, i.e., leaving a single term
in the sums of Eqs. (1)-(3). On the contrary, for the modes with different
azimuthal indices, several harmonics are needed due to weak coupling between
the fields (similar to the TM-TE pair).

Figure 11:  LC symmetry for the hybrid modes EH11-HE21 having different
azimuthal indices.

Figure 12:  LC symmetry for the circularly symmetric TM01, TE01 and hybrid
HE21 modes.

Figure 13:  (a), (b), (c) Electric field $|E_{z}|$ corresponding to the LC-
symmetry thresholds of EH11-HE21, TM01-HE21 and TE01-HE21 mode pairs; (d),
(e), (f) The corresponding magnetic field $|H_{z}|$.

As an example, let us consider the hybrid modes HE21 and EH11. For their
dispersion curves to cross at $f=5.6$ THz and $k_{z}/k_{0}=1.6286$, the radii
of the cylinders should be $R_{1}=11.44$ $\mu$m and $R_{2}=9.91$ $\mu$m.
Figure 13 shows how the number of azimuthal harmonics in Eqs. (1)-(3)
influences the possibility of the loss compensation phenomenon for these
modes. In this figure, $M=2N+1$ is the number of terms took into account in
the field sums. We see that the LC-symmetry threshold is observed for the
values $\tan{\delta_{1}}=6.23\cdot 10^{-4}$ and $\tan{\delta_{2}}=-4.65\cdot
10^{-4}$ at the distance $d=26.01$ $\mu$m between the waveguides only for
$M\geq 5$, i.e., the minimal set of harmonics includes the terms with
$n=\\{0,\pm 1,\pm 2\\}$. This result is confirmed by the comparison with the
independent full-wave calculation using COMSOL Multiphysics® software (see
symbols in Fig. 13). The required number of harmonics becomes larger for very
close cylinders (near $d=0$) in order to save the consistency between the
analytical and numerical methods as demonstrated with the calculations for
$M=11$ in Fig. 13. Further, we omit the comparison with the full-wave
simulations, since we are interested mainly in the region close to the LC-
symmetry thresholds where the consistency is perfect even for relatively small
$M$.

Since the hybrid modes can be coupled to the symmetric TM and TE modes, we
analyze the loss compensation for such situations as well. For the pair
TM01-HE21, for example, the LC-symmetry threshold can be reached at $d=25.58$
$\mu$m, whereas it is $d=28.31$ $\mu$m for TE01-HE21 (see Fig. 13). The
parameters of the system should be taken as follows:

  * •

TM01-HE21 – {$R_{1}=10$ $\mu$m, $R_{2}=9.91$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=5\cdot 10^{-4}$,
$\tan{\delta_{2}}=-3.44\cdot 10^{-4}$};

  * •

TE01-HE21 – {$R_{1}=7.48$ $\mu$m, $R_{2}=9.91$ $\mu$m,
$\epsilon_{r1}=\epsilon_{r2}=12$, $\tan{\delta_{1}}=5\cdot 10^{-4}$,
$\tan{\delta_{2}}=-3.32\cdot 10^{-4}$};

Note that for the TM01-HE21 pair at the LC-symmetry threshold, the
longitudinal electric field $E_{z}$ is excited in both cylinders [Fig. 13(b)],
whereas the longitudinal magnetic field $H_{z}$ exists only in the second one
due to the HE21 mode [Fig. 13(e)]. The opposite is true for the TE01-HE21
pair: $E_{z}$ is absent in the first cylinder [Fig. 13(c)], but $H_{z}$ is
present in both waveguides [Fig. 13(f)].

##  V Conclusion

We employ the multi-mode analytical approach to study the $\mathcal{PT}$ and
LC symmetries in the dielectric cylindrical waveguides with gain and loss. It
is shown that for any predetermined frequency, one can realize the loss
compensation by tuning the waveguides to the proper modes and to the proper
radii. For the modes of either TM or TE nature, we can reach loss compensation
either in the $\mathcal{PT}$-symmetric phase (when the waveguides have
identical radii) or at the LC-symmetry threshold (when the waveguides are
dissimilar). The loss compensation is also possible in the situations
involving the hybrid modes with identical or different azimuthal indices. By
tuning the waveguides radii, we obtained the LC-symmetry thresholds for a wide
number of mode pairs, namely {HE-HE, HE-EH, HE-TM, HE-TE}. The results are
illustrated with the profiles of $E_{z}$ and $H_{z}$ components of
electromagnetic field at the corresponding EPs (LC- and
$\mathcal{PT}$-symmetry thresholds) and corroborated by comparing with the
full-wave simulations.

Our results show that the loss compensation phenomenon in loss-gain systems is
the general effect taking place even in asymmetric systems, such as coupled
dielectric waveguides of different radii. Tuning the waveguides to the desired
modes simply by changing the distance between them and their radii allows to
build the lossless system starting from the arbitrary values of gain and loss.
These results significantly expand the understanding of the concepts of LC and
$\mathcal{PT}$ symmetry and can be applied for the experimental implementation
of new types of optical devices.

###### Acknowledgements.

The work was supported by the National Key R&D Program of China (Project No.
2018YFE0119900) and the State Committee on Science and Technology of Belarus
(Project No. F20KITG-010).

## Appendix: Derivation of transverse field components

The transverse field components $(E_{r},H_{r},E_{\phi},H_{\phi})$ in all
regions can be readily expressed in terms of axial components Eqs. (1)-(3)
from the Maxwell equations. The $\phi$-components of the fields in the first
cylinder (Region I) can be derived as:

$\begin{split}E_{\phi}^{1}&=\frac{1}{k_{p,1}^{2}}\sum_{n}{\left[-\frac{k_{z}n}{r_{1}}A_{n}^{1}J_{n}(k_{p,1}r_{1})-i\omega\mu_{0}B_{n}^{1}k_{p,1}J^{\prime}_{n}(k_{p,1}r_{1})\right]e^{in\phi_{1}}},\\\
H_{\phi}^{1}&=\frac{1}{k_{p,1}^{2}}\sum_{n}{\left[i\omega\epsilon_{1}k_{p,q}A_{n}^{1}J^{\prime}_{n}(k_{p,1}r_{1})-\frac{k_{z}n}{r_{1}}B_{n}^{1}J_{n}(k_{p,1}r_{1})\right]e^{in\phi_{1}}},\end{split}$
(4)

The $\phi$-components of the fields inside the second cylinder (Region II):

$\begin{split}E_{\phi}^{2}&=\frac{1}{k_{p,2}^{2}}\sum_{n}{\left[-\frac{k_{z}n}{r_{2}}A_{n}^{2}J_{n}(k_{p,2}r_{2})-i\omega\mu_{0}B_{n}^{2}k_{p,q}J^{\prime}_{n}(k_{p,2}r_{2})\right]e^{in\phi_{2}}},\\\
H_{\phi}^{2}&=\frac{1}{k_{p,2}^{2}}\sum_{n}{\left[i\omega\epsilon_{2}k_{p,2}A_{n}^{2}J^{\prime}_{n}(k_{p,2}r_{2})-\frac{k_{z}n}{r_{2}}B_{n}^{2}J_{n}(k_{p,2}r_{2})\right]e^{in\phi_{2}}},\end{split}$
(5)

For derivation of the fields in the Region III, we connect the coordinates
$(r_{1},\phi_{1})$ and $(r_{2},\phi_{2})$ using the Graf addition theorem:

$\begin{split}B_{n}(r_{1})e^{\pm
in\phi_{1}}=\sum_{k=-N}^{N}{B_{n+k}(h)J_{k}(r_{2})e^{\mp ik\phi_{2}}}e^{\pm
ik\pi},\\\ B_{n}(r_{2})e^{\pm
in\phi_{2}}=\sum_{k=-N}^{N}{B_{n+k}(h)J_{k}(r_{1})e^{\mp ik\phi_{1}}}e^{\pm
in\pi},\end{split}$ (6)

where $B_{n}(\cdot)$ is the $n$-th order cylindrical function and $h$ is the
distance between the centers of two cylinders ($h=d+R_{1}+R_{2}$) (see Fig.
1). The $\phi$-components of the fields outside the cylinders in coordinates
$(r_{1},\phi_{1})$ are:

$\begin{split}E_{\phi}^{3}(r_{1},\phi_{1})=&\frac{1}{k_{p,3}^{2}}\sum_{n}\left\\{-\frac{k_{z}n}{r_{1}}\left[C_{n}^{1}H_{n}^{(1,2)}(k_{p,3}r_{1})e^{in\phi_{1}}+\sum_{k}{C_{n}^{2}H_{n+k}^{(1,2)}(k_{p,3}h)J_{k}(k_{p,3}r_{1})e^{-ik\phi_{1}}e^{in\pi}}\right]-\right.\\\
&\left.-i\omega\mu_{0}B_{n}^{2}k_{p,3}\left[D_{n}^{1}H_{n}^{{}^{\prime}(1,2)}(k_{p,3}r_{1})e^{in\phi_{1}}+\sum_{k}{D_{n}^{2}H_{n+k}^{(1,2)}(k_{p,3}h)J^{\prime}_{k}(k_{p,3}r_{1})e^{-ik\phi_{1}}e^{in\pi}}\right]\right\\},\\\
H_{\phi}^{3}(r_{1},\phi_{1})=&\frac{1}{k_{p,3}^{2}}\sum_{n}\left\\{i\omega\epsilon_{3}k_{p,3}\left[C_{n}^{1}H_{n}^{{}^{\prime}(1,2)}(k_{p,3}r_{1})e^{in\phi_{1}}+\sum_{k}{C_{n}^{2}H_{n+k}^{(1,2)}(k_{p,3}h)J^{\prime}_{k}(k_{p,3}r_{1})e^{-ik\phi_{1}}e^{in\pi}}\right]\right.-\\\
&-\left.\frac{k_{z}n}{r_{1}}\left[D_{n}^{1}H_{n}^{(1,2)}(k_{p,3}r_{1})e^{in\phi_{1}}+\sum_{k}{D_{n}^{2}H_{n+k}^{(1,2)}(k_{p,3}h)J_{k}(k_{p,3}r_{1})e^{-ik\phi_{1}}e^{in\pi}}\right]\right\\},\end{split}$
(7)

The $\phi$-components of the fields in the Region III in the coordinates
$(r_{2},\phi_{2})$ are:

$\begin{split}E_{\phi}^{3}(r_{2},\phi_{2})=&\frac{1}{k_{p,3}^{2}}\sum_{n}\left\\{-\frac{k_{z}n}{r_{2}}\left[\sum_{k}{C_{n}^{1}H_{n+k}^{(1,2)}(k_{p,3}h)J_{k}(k_{p,3}r_{2})e^{ik(\pi-\phi_{2})}}+C_{n}^{2}H_{n}^{(1,2)}(k_{p,3}r_{2})e^{in\phi_{2}}\right]-\right.\\\
&\left.-i\omega\mu_{0}k_{p,3}\left[\sum_{k}{D_{n}^{1}H_{n+k}^{(1,2)}(k_{p,3}h)J^{\prime}_{k}(k_{p,3}r_{2})e^{ik(\pi-\phi_{2})}}+D_{n}^{2}H_{n}^{{}^{\prime}(1,2)}(k_{p,3}r_{2})e^{in\phi_{2}}\right]\right\\},\\\
H_{\phi}^{3}(r_{2},\phi_{2})=&\frac{1}{k_{p,3}^{2}}\sum_{n}\left\\{i\omega\epsilon_{3}k_{p,3}\left[\sum_{k}{C_{n}^{1}H_{n+k}^{(1,2)}(k_{p,3}h)J^{\prime}_{k}(k_{p,3}r_{2})e^{ik(\pi-\phi_{2})}}+C_{n}^{2}H_{n}^{{}^{\prime}(1,2)}(k_{p,3}r_{2})e^{in\phi_{2}}\right]\right.-\\\
&-\left.\frac{k_{z}n}{r_{2}}\left[\sum_{k}{D_{n}^{1}H_{n+k}^{(1,2)}(k_{p,3}h)J_{k}(k_{p,3}r_{2})e^{ik(\pi-\phi_{2})}}+D_{n}^{2}H_{n}^{(1,2)}(k_{p,3}r_{2})e^{in\phi_{2}}\right]\right\\},\end{split}$
(8)

The explicit form of the dispersion relation is the condition of zero
determinant of the following system of equations:

$\begin{split}\tilde{A}_{m}^{1}J_{m}(k_{p,1}r_{1})=\tilde{C}_{m}^{1}H_{m}^{(1,2)}(k_{p,3}r_{1})+\sum_{n}{\tilde{C}_{n}^{2}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{1})},\\\
B_{m}^{1}J_{m}(k_{p,1}r_{1})=D_{m}^{1}H_{m}^{(1,2)}(k_{p,3}r_{1})+\sum_{n}{D_{n}^{2}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{1})},\\\
\tilde{A}_{m}^{2}J_{m}(k_{p,2}r_{2})=\sum_{n}{\tilde{C}_{n}^{1}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{2})}+\tilde{C}_{m}^{2}H_{m}^{(1,2)}(k_{p,3}r_{2}),\\\
B_{m}^{2}J_{m}(k_{p,2}r_{2})=\sum_{n}{D_{n}^{1}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{2})}+D_{m}^{2}H_{m}^{(1,2)}(k_{p,3}r_{2}),\\\
\frac{1}{k_{p,1}^{2}}\left[-\frac{k_{z}m}{r_{1}}\tilde{A}_{m}^{1}J_{m}(k_{p,1}r_{1})-ik_{0}B_{m}^{1}k_{p,1}J^{\prime}_{m}(k_{p,1}r_{1})\right]=\\\
=\frac{1}{k_{p,3}^{2}}\left\\{-\frac{k_{z}m}{r_{1}}\left[\tilde{C}_{m}^{1}H_{m}^{(1,2)}(k_{p,3}r_{1})+\sum_{n}{\tilde{C}_{n}^{2}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{1})}\right]\right.-\\\
-\left.ik_{0}k_{p,3}\left[D_{m}^{1}H_{m}^{{}^{\prime}(1,2)}(k_{p,3}r_{1})+\sum_{n}{D_{n}^{2}}H_{m-n}^{(1,2)}(k_{p,3}h)J^{\prime}_{m}(k_{p,3}r_{1})\right]\right\\},\\\
\frac{1}{k_{p,1}^{2}}\left[ik_{0}\epsilon_{1}k_{p,1}\tilde{A}_{m}^{1}J^{\prime}_{m}(k_{p,1}r_{1})-\frac{k_{z}m}{r_{1}}B_{m}^{1}J_{m}(k_{p,1}r_{1})\right]=\\\
=\frac{1}{k_{p,3}^{2}}\left\\{ik_{0}\epsilon_{3}k_{p,3}\left[\tilde{C}_{m}^{1}H_{m}^{{}^{\prime}(1,2)}(k_{p,3}r_{1})+\sum_{n}{\tilde{C}_{n}^{2}H_{m-n}^{(1,2)}(k_{p,3}h)J^{\prime}_{m}(k_{p,3}r_{1})}\right]\right.-\\\
-\left.\frac{k_{z}m}{r_{1}}\left[D_{m}^{1}H_{m}^{(1,2)}(k_{p,3}r_{1})+\sum_{n}{D_{n}^{2}}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{1})\right]\right\\},\\\
\frac{1}{k_{p,2}^{2}}\left[-\frac{k_{z}m}{r_{2}}\tilde{A}_{m}^{2}J_{m}(k_{p,2}r_{2})-ik_{0}B_{m}^{2}k_{p,2}J^{\prime}_{m}(k_{p,2}r_{2})\right]=\\\
=\frac{1}{k_{p,3}^{2}}\left\\{-\frac{k_{z}m}{r_{2}}\left[\sum_{n}{\tilde{C}_{n}^{1}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{2})}+\tilde{C}_{m}^{2}H_{m}^{(1,2)}(k_{p,3}r_{2})\right]\right.-\\\
-\left.ik_{0}k_{p,3}\left[\sum_{n}{D_{n}^{1}}H_{m-n}^{(1,2)}(k_{p,3}h)J^{\prime}_{m}(k_{p,3}r_{2})+D_{m}^{2}H_{m}^{{}^{\prime}(1,2)}(k_{p,3}r_{2})\right]\right\\},\\\
\frac{1}{k_{p,2}^{2}}\left[ik_{0}\epsilon_{2}k_{p,2}\tilde{A}_{m}^{2}J^{\prime}_{m}(k_{p,2}r_{2})-\frac{k_{z}m}{r_{2}}B_{m}^{2}J_{m}(k_{p,2}r_{2})\right]=\\\
=\frac{1}{k_{p,3}^{2}}\left\\{ik_{0}\epsilon_{3}k_{p,3}\left[\sum_{n}{\tilde{C}_{n}^{1}H_{m-n}^{(1,2)}(k_{p,3}h)J^{\prime}_{m}(k_{p,3}r_{2})}+\tilde{C}_{m}^{2}H_{m}^{{}^{\prime}(1,2)}(k_{p,3}r_{2})\right]\right.-\\\
-\left.\frac{k_{z}m}{r_{2}}\left[\sum_{n}{D_{n}^{1}}H_{m-n}^{(1,2)}(k_{p,3}h)J_{m}(k_{p,3}r_{2})+D_{m}^{2}H_{m}^{(1,2)}(k_{p,3}r_{2})\right]\right\\}.\end{split}$
(9)

Here $\tilde{A}_{n}^{1,2}=\sqrt{\epsilon_{0}/\mu_{0}}A_{n}^{1,2}$,
$\tilde{C}_{n}^{1,2}=\sqrt{\epsilon_{0}/\mu_{0}}C_{n}^{1,2}$.

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