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Topological Phases in Ring Resonators: Recent Progress and Future Prospects Techniques to Their Limits

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Topological Phases in Ring Resonators: Recent Progress and Future Prospects Techniques to Their Limits Nanophotonics 2020; 9(15): 4473–4487 Review Daniel Leykam* and Luqi Yuan* Topological phases in ring resonators: recent progress and future prospects https://doi.org/10.1515/nanoph-2020-0415 techniques to their limits. Continued progress will require Received July 23, 2020; accepted September 12, 2020; new approaches to minimize the detrimental influence of published online September 25, 2020 fabrication imperfections and disorder. Topological pho- tonics is a young subfield of photonics that seeks to Abstract: Topological photonics has emerged as a novel address this challenge using novel design approaches paradigm for the design of electromagnetic systems from inspired by exotic electronic condensed matter materials microwaves to nanophotonics. Studies to date have largely such as topological insulators [1, 2]. Loosely speaking, to- focused on the demonstration of fundamental concepts, pological systems provide a systematic way to create such as nonreciprocity and waveguiding protected against disorder-robust modes or observables using a collection of fabrication disorder. Moving forward, there is a pressing imperfect components or modes. For example, certain need to identify applications where topological designs classes of “topologically nontrivial” systems exhibit spe- can lead to useful improvements in device performance. cial edge modes that can propagate reliably without Here, we review applications of topological photonics to backscattering even in the presence of strongly scattering ring resonator–based systems, including one- and two- defects, forming the basis for superior optical waveguides. dimensional resonator arrays, and dynamically modulated One natural setting where this robustness can poten- resonators. We evaluate potential applications such as tially be useful is in the design of integrated photonic cir- quantum light generation, disorder-robust delay lines, and cuits [3], where strong light confinement brings sensitivity to optical isolation, as well as future research directions and nanometre-scale fabrication imperfections. However, there open problems that need to be addressed. is not yet any disruptive killer application where topological Keywords: coupled resonator optical waveguide; optical photonic devices have achieved superior performance isolator; ring resonator; silicon photonics; topological compared to mature design paradigms, despite growing photonics. interest in topological photonics since seminal works pub- lished in 2008 [4, 5]. To help bridge this gap, several reviews have been published recently, some providing compre- 1 Introduction hensive surveys of topological photonic systems [1, 2, 6, 7], and others focussing on specificapplicationssuchas Demand for miniaturized optical components such as incorporating topological concepts into active devices such waveguides and lenses that can be incorporated into as lasers [8], nanophotonics [9], nonreciprocal devices [10], compact photonic devices is pushing fabrication and nonlinear optical processes [11]. The aim of this brief review is to complement these Present address: Daniel Leykam, Centre for Quantum Technologies, recent surveys with a concise introduction to applications National University of Singapore, 3 Science Drive 2, Singapore 117543, of ideas from topological photonics to optical ring reso- Singapore. nator–based systems. Ring resonators are a versatile and important ingredient of integrated photonic circuits as *Corresponding authors: Daniel Leykam, Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon, they can be used as compact filters, sensors, and delay 34126, Korea; and Basic Science Program, Korea University of Science lines and as a means of enhancing nonlinear optical ef- and Technology, Daejeon, 34113, Korea, fects [12]. However, active tuning is typically required to E-mail: [email protected]. https://orcid.org/0000-0002- compensate for resonance shifts induced by various per- 8588-9886; and Luqi Yuan, State Key Laboratory of Advanced Optical turbations, increasing device complexity and energy Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China, consumption [13, 14]. We will discuss some of the ways in E-mail: [email protected]. https://orcid.org/0000-0001-9481- which topological designs may lead to superior devices 0247 with improved reliability. We will focus on topological Open Access. © 2020 Daniel Leykam and Luqi Yuan, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 4474 D. Leykam and L. Yuan: Topological phases in ring resonators systems formed by coupling together multiple resonators the latter require a combination of time-reversal symmetry to form a lattice or by considering coupling between and some other internal symmetry protecting a spin-like multiple resonances of a single ring using external mod- degree of freedom (known as quantum spin-Hall topolog- ulation or nonlinearity. In both cases, the lattice formed by ical phases [16]). In both cases, the resulting topological the coupled modes can be designed to have topological edge states are of interest as a means of creating disorder- band structures and achieve protection against certain robust optical waveguides. kinds of disorder. The generic approach to create a topological system is The outline of this article is as follows: Section 2 starts to start with a simple periodic medium exhibiting some with a brief overview of the basic concepts underlying to- degeneracy in its photonic band structure and then break pological photonics and ring resonators. Next, we review one of the symmetries protecting the degeneracy to open a the implementation of topological photonics using arrays band gap. Breaking the symmetry in the right way will of coupled ring resonators in Section 3. Section 4 discusses create a topologically nontrivial band gap hosting pro- how we can use dynamic modulation as a novel degree of tected edge modes. freedom for the engineering of topological effects in ring Figure 1 shows a simple one-dimensional example of resonators. We discuss future research directions and this idea, where one can reduce the translation symmetry promising potential applications of topological ring reso- of a periodic lattice by staggering its site positions to create nators in Section 5, before concluding with Section 6. a lattice of dimers. Reducing the translation symmetry opens a minigap the lattice’s band structure, analogous to the Su–Schrieffer–Heeger tight binding Hamiltonian for 2 Background electron transport in polymers [17, 18], H ∑J a† a + J a† a + . ., SSH 1 2n−1 2n 2 2n 2n+1 c c (1) 2.1 Topological photonics n a n J The key idea underlying topological photonics is the bulk- where n is the annihilation operator for the th site and 1,2 H boundary correspondence, which states that the topolog- are coupling strengths. Note that while SSH is written us- ical properties of the photonic band structure of a bulk ing second quantization notation, it is equally applicable H periodic medium can be related to the appearance of robust to classical states of light, where the eigenvalues of SSH modes localized to edges or domain walls of the system give the frequency detunings of the collective array modes, [1, 2]. These “topologically protected” edge modes have as shown in Figure 1. very different properties compared to conventional defect There are two different ways to dimerize a finite lattice; J J J J modes. For example, as they arise owing to the topological we can have either 1 > 2 or 1 < 2, corresponding to two properties of the bulk photonic band structure, they are inequivalent topological phases distinguished by a quan- J J robust against certain classes of local perturbations at the tized winding number. The trivial phase 1 > 2 forms a set of edge or domain wall and can only be removed by large dimers and does not exhibit any end modes. On the other perturbations capable of closing the bulk band gap. Thus, hand, in the nontrivial phase J1 < J2, the dimers are broken topology allows us to create special robust boundary at the ends of the system, resulting in a pair of localized end modes protected by a higher dimensional bulk. modes with frequencies lying in the middle of the band For the purposes of this review, two classes of topo- gap. The end modes are topologically protected in the logical modes are of interest. One-dimensional media can sense that introducing disorder to the intersite couplings exhibit topologically protected modes localized to the ends does not shift their frequencies; they remain “pinned” to of the system. These modes have their frequencies pinned the middle of the band gap and localized to the ends as to the middle of the band gap, even in the presence of long as the disorder is sufficiently weak that the bulk band disorder, as long as certain symmetries are preserved. In gap remains open. this case, topology provides a systematic way to create Thanks to its simplicity and relative ease of imple- localized defect modes at a specific frequency. mentation, photonic analogues of the Su–Schrieffer– The second important class of topological modes is Heeger model have been realized in a variety of platforms edge states of two-dimensional topological systems. These [1], including waveguide lattices [19–21], plasmonic and exhibit either unidirectional or spin-controlled propaga- dielectric nanoparticle arrays [22, 23], photonic crystals tion along
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