Necklaces of PT-symmetric dimers
D. J. NODAL STEVENS,
1
BENJAMÍN JARAMILLO ÁVILA,
2
AND B. M. RODRÍGUEZ-LARA
1,3,
*
1
Tecnologico de Monterrey, Escuela de Ingenierı´a y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L. 64849, Mexico
2
CONACYT–Instituto Nacional de Astrofı´sica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico
3
Instituto Nacional de Astrofı´sica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico
*Corresponding author: bmlara@itesm.mx
Received 5 September 2017; revised 25 January 2018; accepted 9 February 2018; posted 9 February 2018 (Doc. ID 306303);
published 4 April 2018
We study light propagation through cyclic arrays, composed by copies of a given PT-symmetric dimer, using a
group theoretical approach and finite element modeling. The theoretical mode-coupling analysis suggests the use
of these devices as output port replicators where the dynamics is controlled by the impinging light field. This is
confirmed in good agreement with finite element propagation in an experimentally feasible necklace of passive
PT-symmetric dimers constructed from lossy and lossless waveguides.
© 2018 Chinese Laser Press
OCIS codes: (230.7370) Waveguides; (130.2790) Guided waves; (140.4480) Optical amplifiers; (000.1600) Classical and
quantum physics.
https://doi.org/10.1364/PRJ.6.000A31
1. INTRODUCTION
Photonic device designers have been toying with the idea of
using gain and loss in linear [1] and nonlinear [2] optical sys-
tems in order to produce unidirectional couplers or low inten-
sity switches, in that order, for a long time. The advent of parity
time (PT) symmetry in quantum mechanics [3,4], the idea that
non-Hermitian Hamiltonians invariant under space–time re-
flection might possess a real spectrum, brought a new structure
to these approaches. Two ideas made use of this quantum
mechanical tool to describe a planar slab waveguide [5] and
coupled optical structures with symmetric gain and losses
[6]. The latter became seminal for the field of PT-symmetric
photonics and has inspired a plethora of optical proposals [7,8].
The quintessential PT-symmetric optical device is the
balanced gain and loss dimer that has been experimentally real-
ized in diverse optical platforms [9–13]. It can be understood as
a finite, nonunitary realization of the Lorentz group [14]
allowing for three types of propagation dynamics: periodic field
amplitude oscillation with amplification, as well as linear and
exponential field amplitude amplification. The first case corre-
sponds to the PT-symmetric regime, where eigenvalues are real.
The second case is the Kato exceptional point, where the
eigenvalues are degenerate and equal to zero. The third case
displays PT-symmetry breaking, where the two eigenvalues
are purely imaginary and the complex conjugates of each other.
It is also known that any optical realization of the Lorentz
group in the broken PT-symmetric regime shows asymptotic
behavior, for renormalized intensities, that depends only on
the interplay of gain and coupling parameters for both linear
[14] and nonlinear systems [7].
We are interested in a natural extension of the nonlinear
PT-symmetric dimer: its periodic repetition over a circular
loop. Light propagation through these so-called necklaces
where individual waveguides are homogeneously spaced has
been shown to (dis)allow a PT-symmetric regime for (even)
odd repetition of the dimer [15]. The nonlinear four-waveguide
array is known to produce an asymmetric distribution of optical
power [16], to possess continuous families of nonlinear modes
[17], and it is not PT symmetric in the usual matrix sense [18].
The absence of exceptional points, for homogeneous arrays of
dimension four, was discussed using linear arrays of four sub-
wavelength waveguides [19], while the possibility to restore PT
symmetry through mechanical action has also been shown [20].
Here, we will construct a slightly more general system where
the standard PT-symmetric dimer is repeated for N times over
a circular loop, with the addition of constant intra- and inter-
dimer coupling lengths that are different from each other, as
shown in Fig. 1(a). Such a model can be realized in laser in-
scribed arrays of waveguides, circular multicore fibers, toroidal
cavities, or electronic systems. In the following sections, we in-
troduce the model and show that it can be composed using the
cyclic and Lorentz groups. We will use a group theory approach
to further the idea that discrete symmetries in photonics are a
powerful design tool [21]. Then, we diagonal ize our coupled
mode model in the cyclic group basis to show that its dynamics
is consistent with those of uncoupled effective dimers with var-
iable couplings. We discuss the eigenvalues of these effective
dimers and show that homogeneously distributed necklaces
with even numbers of copies always have at least a pair of imagi-
nary eigenvalues and PT symmetry can be restored by making
Research Article
Vol. 6, No. 5 / May 2018 / Photonics Research A31
2327-9125/18/050A31-07 Journal © 2018 Chinese Laser Press
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