A simple tight-binding model to study the band-structure of an infinite length Parity-time (PT) symmetric chain of resonators is presented in this paper. In the talk, we will investigate the impact of having a structure of finite length and consider the effect of a modulation of the real part of the refractive index on the complete band-structure of the PT-chain system. For a finite PT-chain structure under certain modulation, we observe the existence of a localised mode which is either lasing or dissipatin

Benson, Trevor M.

Creagh, Stephen

Gradoni, Gabriele

Phang, Sendy

Sewell, Phillip

Vukovic, Ana

Nottingham eTheses

Parity-time symmetric chain resonators

Name not available

Nottingham ePrints

Phang, Sendy and Vukovic, Ana and Creagh, Stephen and Gradoni, Gabriele and Sewell, Phillip and Benson, Trevor M. (2016) Parity-time symmetric chain resonators. In: 7th International Conference on Metamaterials, Photonic Crystals and Plasmonics (META 16), 25-28 July 2016, Malaga, Spain. Access from the University of Nottingham repository: http://eprints.nottingham.ac.uk/35641/1/META16.pdfCopyright and reuse: The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions.This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence.  For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdfA note on versions: The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk            Parity-Time Symmetric Chain Resonators   S. Phang1,2, A. Vukovic1, S. C. Creagh2, G. Gradoni2, P. Sewell1 and T. M. Benson1 1George Green Institute for Electromagnetics Research, University of Nottingham, UK ‎2School of Mathematical Sciences, University of Nottingham, UK *corresponding author: sendy.phang@nottingham.ac.uk   Abstract- A simple tight-binding model to study the band-structure of an infinite length Parity-time (PT) symmetric chain of resonators is presented in this paper. In the talk, we will investigate the impact of having a structure of finite length and consider the effect of a modulation of the real part of the refractive index on the complete band-structure of the PT-chain system. For a finite PT-chain structure under certain modulation, we observe the existence of a localised mode which is either lasing or dissipating.    Photonic structures with combined gain and loss, known as Parity-Time (PT) symmetric structures, have been shown to exhibit unique properties [1,2]. Within the context of PT-resonator structures, we have reported an analytical approach for calculating the threshold point of such structures using a Boundary Integral Equation method[1]. Furthermore we also analysed the impact of unbalancing gain/loss in the PT-resonator structure in which we observed a laser operation in a loss dominated structure [1]. In this contribution, we report our recent study of a PT-symmetric chain of resonator structures (see Fig. 1).   Figure 1. Schematic illustration of the PT-chain resonator structure. The PT-chain considered is schematically illustrated in Fig. 1; each cell is comprised of 4 cylindrical resonators of the same radius 𝑟 and separated by a fixed gap 𝑔. The refractive index within a unit cell is given by:  {  𝑛1 = 𝑛0 + Δ𝑛′ + 𝑗𝑛′′𝑛2 = 𝑛0 − Δ𝑛′ + 𝑗𝑛′′𝑛3 = 𝑛0 − Δ𝑛′ − 𝑗𝑛′′𝑛4 = 𝑛0 + Δ𝑛′ − 𝑗𝑛′′ (1) where, 𝑛0, Δ𝑛′ and 𝑛′′ are the average refractive index, modulation of the real part of the refractive index and modulation of the imaginary part of the refractive index (gain/loss), respectively.   In this work, we investigate the impact of having a structure of finite length and consider the effect of a modulation of the real part of the refractive index on the complete band-structure of the PT-chain system. A simple tight-binding model of the infinite chain for the case of no real modulation, i.e. Δ𝑛′ = 0 is given by, 𝐻 =(  𝜔0 𝜅 𝑒𝑗𝛽Λ𝜅𝜅 𝜔0 𝜅𝜅 𝜔0∗ 𝜅𝑒−𝑗𝛽Λ𝜅 𝜅 𝜔0∗)   (2) where 𝜔0 is the complex resonant frequency of the resonator, 𝜅 is the coupling strength between resonators and 𝛽Λ is the Bloch phase. The Hamiltonian 𝐻 has multiple degeneracies even when the non-Hermitian parameter,  𝛾 = ℑ(𝜔0)/𝜅 is zero, see Fig. 2(a); thus a PT-chain structure without modulation Δ𝑛′ = 0 will spontaneously break the PT-symmetry for a non-zero 𝛾, i.e. 𝑛′′ > 0, as shown in Fig. 2(b).         Figure 2. Band-structure of an infinitely long PT-chain with no real part modulation Δn′ = 0  Meanwhile, by introducing a small real part modulation Δ𝑛′ ≠ 0, the resonant frequencies of the resonator are shifted, i.e. they decrease if Δ𝑛′ < 0 and increase if Δ𝑛′ > 0; now the Hamiltonian is given by,  𝐻 =(  𝜔1 𝜅 𝑒𝑗𝛽Λ𝜅𝜅 𝜔2 𝜅𝜅 𝜔2∗ 𝜅𝑒−𝑗𝛽Λ𝜅 𝜅 𝜔1∗)   (3) where 𝜔1 −𝜔2 = 2δ and 𝜔0 = (𝜔1 +𝜔2)/2. The real part and the imaginary part of the eigenfrequencies of (3) are plotted in Fig. 3(a,b) respectively for 𝛿 = 0.25𝜔0. It can be seen in Fig. 3(a) that by introducing real modulation Δ𝑛′, the eigenfrequencies at the edge of the Brillouin zone are no longer degenerate. As such there exists a minimum threshold of the 𝛾 parameter which causes PT-symmetry breaking.   Figure3. Band-structure of an infinitely long PT-chain with real modulation  In the talk, we will discuss the effect of having a finite length of chain in which we observed the presence of localised broken-PT-symmetry mode, which has a characteristic of either lasing or decaying and is localised at either end of the chain structure.  REFERENCES 1. Sendy Phang, Ana Vukovic, Stephen C. Creagh, Trevor M. Benson, Phillip D. Sewell, and Gabriele Gradoni, "Parity-time symmetric coupled microresonators with a dispersive gain/loss," Opt. Express 23, 11493-11507 (2015). 2. Sendy Phang, Ana Vukovic, Stephen C. Creagh, Trevor M. Benson, Phillip D. Sewell, and Gabriele Gradoni, " Localized Single Frequency Lasing States in a Finite Parity-Time Symmetric Resonator Chain," Scientific Reports 6, 20499 (2016). γ (a) (b) (a) (b) 

Parity-time symmetric chain resonators

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