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Necklaces of PT-symmetric dimers

D. J. NODAL STEVENS,

BENJAMÍN JARAMILLO ÁVILA,

AND B. M. RODRÍGUEZ-LARA

1,3,

Tecnologico de Monterrey, Escuela de Ingenierı´a y Ciencias, Ave. Eugenio Garza Sada 2501, Monterrey, N.L. 64849, Mexico

CONACYT–Instituto Nacional de Astrofı´sica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico

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.

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

the intra- and inter-dimer couplings different. Then, we show

that the underlying cyclic symmetry suggests the use of these

photonic devices as output replicators, where input light is used

to select the effective dynamics to replicate in the outputs. We

provide finite element simulations to justify our analytic ap-

proach using data from experimental work on waveguide laser

inscription in silicon. We close our article with a brief summary

and conclusion.

2. LINEAR MODEL AND ITS UNDERLYING

SYMMETRY

The coupled mode formalism for a tightly bound necklace

formed by N identical PT-symmetric dimers yields the follow-

ing set of differential equations:

−i∂

ziγE

zg

−iϕ

zg

iϕ

2N −1

z;

−i∂

z−iγE

zg

iϕ

zg

−iϕ

z;

−i∂

ziγE

zg

−iϕ

2j1

zg

iϕ

2j−1

z;

−i∂

2j1

z−iγE

2j1

zg

iϕ

zg

−iϕ

2j2

z;

− i∂

2N −2

ziγE

2N −2

zg

−iϕ

2N −1

zg

iϕ

2N −3

z;

−i∂

2N −1

z−iγE

2N −1

zg

iϕ

2N −2

zg

−iϕ

z;

(1)

for the field mode amplitude at each fiber, described by the

complex-valued functions E

z with n  0; …; 2N − 1.

Our optical system generalizes a previous homogeneously

coupled periodic array studied in Ref. [15] by allowing different

intra(inter)-dimer couplings, g

−iϕ

), where the cou-

pling strength, g

), and coupling phases, ϕ

(ϕ

), are real.

The coupling phases can be induced by mechanical twisting in

optical fibers or capacitive coupling in electronic systems.

Finally, gains and losses in the fibers are quantified by the real

parameter γ. Our periodic array of coupled waveguides is

sketched in Fig. 1(a) for the general necklace; we also show

sketches for specific realization like the standard PT-symmetric

dimer, Fig. 1(b), the PT-symmetric plaquette, Fig. 1(c), and a

necklace composed by three dimers, Fig. 1(d). Later, we will

focus on the last two of these realizations.

The coupled mode equations in Eq. (1) can be written in a

more compact way using Dirac notation,

−i∂

jEzi 

HjEzi; (2)

where the ket jEzi is a simple vector collecting the 2N field

mode amplitudes in our model,

jEzi 

z

2N −2

z

2N −1

z



j  0 − th dimer

k  0

k  1



j  1 − th dimer

k  0

k  1



j  N − 1 − th dimer

k  0

k  1

(3)

The mode-coupling matrix is represented by the operator

H, a block-tridiagonal 2N × 2N matrix which can be written

using N × N blocks of 2 × 2 matrices,

H 

−

…



−

…



−

0 …



; (4)

where the symbol

0 stands for the 2 × 2 zero matrix, and the

auxiliary block matrices,

Fig. 1. Sketch for necklaces formed by the repetition of identical

PT-symmetric dimers. (a) Necklace where intradimer couplings are

larger than interdimer couplings, g

, and repetition factor N.

(b) Standard PT-symmetric dimer, N  1; (c) PT-symmetric

plaquette, N  2; and (d) necklace with repetition factor N  3.

A32 Vol. 6, No. 5 / May 2018 / Photonics Research

Research Article

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