PT-symmetric dimers项链：光控制的输出端复制器

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本文主要探讨了光在PT对称二元结构项链（Necklaces of PT-symmetric dimers）中的传播特性。PT对称性是一种非Hermitian量子力学的概念，它结合了空间反演（P）和时间反转（T）对称性，这种对称性在光子学领域引起了广泛兴趣，因为它允许设计出具有独特光学性质的结构。研究者采用了一种群论方法与有限元模拟相结合的技术，对由相同PT对称单元构成的循环阵列进行了深入分析。通过理论上的模式耦合分析，他们揭示了这些系统作为光场控制输出端口复制器的潜力。在这种装置中，入射光场能够精确调控光的传输行为和动态特性。这与实际构建的实验性被动PT对称二元结构项链的数值模拟结果高度吻合，证明了这一理论的有效性。实验项链是由有损和无损元件组成的，这种设计考虑到了现实中的制造限制和技术可行性。在实验条件下，观察到的光传播模式证实了理论预测，即PT对称二元结构项链能够作为稳定的光路复用器，其性能受输入光的控制，这对于光通信、量子信息处理以及精密光学设备的设计具有重要意义。此外，论文作者还讨论了这项工作的潜在应用，包括但不限于光信号的稳定传输、模式选择、以及可能的自适应光学功能。由于其独特的对称性和控制能力，这种结构可能为未来的光子学器件提供新的设计思路。总结来说，本文的研究不仅深化了我们对光在PT对称结构中的行为理解，还展示了如何通过设计巧妙的循环阵列来实现对光传输的精细操控。这为未来的光子学研究和应用开发提供了重要的理论支持和技术路径。

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

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