## Abstract

We review recent advances of topological photonics in coupled waveguide systems. To this end, we discuss the various prevalent platforms, as well as various implementations in one- and two-dimensional systems, including driven, static, nonlinear and quantum systems.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Topological ideas in photonics are rooted in notions from condensed matter physics, in particular those associated with topological insulators [1–4]. This new phase of matter is insulating in the bulk but exhibits conduction along their surface without dissipation or back-scattering, even in the presence of defects and disorder. The field of topological photonics was initiated in 2008 by Haldane and Raghu [5,6], who transferred the key features of the electronic Quantum Hall effect to light waves by proposing a photonic analogue of the anomalous quantum Hall effect. The groundbreaking ideas were subsequently implemented in a photonic crystal setting [7,8], where topological propagation of microwaves along the edge of the system was shown, in essence forming a kind of scatter-free waveguide.

These achievements attracted major interest in the photonics community, as in ordinary waveguides, scattering and back-reflection are a major source of unwanted loss that hinders large-scale optical integration [9,10]. The demonstration of scatter-free light transport along the edge of a photonic system proved the existence of a unidirectional waveguiding mechanism based on topology, such that electromagnetic waves can propagate without back-reflection even in the presence of defect and disorder. This is an ideal transport property, which is unprecedented in photonics. Topological photonics promises unique and robust designs for new photonic device functionalities by providing immunity to performance degradation induced by fabrication imperfections or environmental changes.

A natural platform for topological photonics are integrated coupled waveguide systems, which can be arranged in a periodic fashion [11] in order to cause a band structure similar to that of photonic crystals [12] and electronic systems [13]. Topological photonics in integrated waveguide structures is a flourishing field and provides the basis for numerous innovative applications. In this work, we review recent achievements in the field of waveguide-based topological photonics. The manuscript is organized as follows: We start by summarizing a few fundamental concepts of topology in periodic systems in section 2, with a particular emphasis on topological invariants and the famous bulk-edge correspondence. In section 3, we discuss the various waveguide platforms in terms of advantages and challenges. Section 4 deals with one-dimensional (1D) systems, whereas in section 5, recent achievements in two-dimensional (2D) systems are reviewed. A short outlook into topology in nonlinear and quantum systems is given in section 6. The work is concluded by an outlook.

## 2. Topology in photonics

In solid state physics the field of topology compromises quantized phenomena based on band models described by the mathematical tools of differential geometry [4]. Since band structures are a product of Bloch’s theorem for periodic lattices [14], the topology itself originates from the wave nature of quantum mechanics. The existence of specific symmetries, like particle-hole or time-reversal symmetry (TRS), or their crucial absence, creates non-trivial topologies [15]. In detail, topological models describing wave function with geometrical phases, which are additional phases in cyclic settings [16–18]. However, the requirement of equipping wave propagation with symmetry properties and associated geometrical phases is not only restricted to solid state physics. Due to this fact, in the recent years the field of topological physics has been expanded to microwave systems [8], matter waves [19], acoustics [20], mechanical waves [21], electronic circuits [22] and even photonics. Photonic waveguide arrays provide a versatile platform to observe non-trivial topological effects. Such optical lattices are constructed by a discrete pattern of locally increased refractive indices, called waveguides. Typically, each lattice site hosts a tightly-bound mode which interacts with the adjacent waveguides via evanescent coupling. The light dynamics of such an optical array is governed by a coupled mode equation. An additional periodicity of the waveguide lattice enables the prominent photonic band structure, which relates the transversal and longitudinal wavenumbers to one another. Furthermore, coupled waveguide structures are excellently suitable to selectively modify various underlying symmetries. Especially, the breaking of the TRS, firstly suggested in 1988 by Haldane [23], yields to photonic versions of topological insulators [5,6,24]. The influence of broken TRS can be studied by the introduction of artificial gauge fields, which have been develop as a versatile tool to reveal a plethora of topological effects in photonic systems [25,26].

In fact, a photonic array can be analyzed in the framework of mathematical topology, which was originally investigated by Gauß and Bonnet and has been improved to the so called Chern-Simons theory [27–33]. The seminal work [34] explained, besides other geometrical explanations [35,36], the integer quantum Hall effect [37] and simultaneously introduced topology in the field of physics. The mathematical argumentations in the latter works are directly applicable to the optical settings considered in this review.

Similar to solid state systems, the dynamics in photonic arrays can be described by a real-space Hamiltonian $H$, due to a mathematical equivalence of the Schrödinger equation and the paraxial Helmholtz equation. The main requirement of topological systems is the existence of a band gap in the spectrum of the corresponding Bulk Hamiltonian $H_{\mathrm {B}}$, obtained by a Bloch approach of the infinitely extended structure $H$. Such band structures are easily achievable in periodic photonic structures [38]. Generally, the bulk Hamiltonian provides the objects, which are treated by the tools of mathematical topology. However, the physical setting itself determines the appropriate topological analysis. Even in the works presented in this review various approaches for measuring the underlying topology, i.e. different topological invariants, are used. Therefore, a complete theoretical description exceeds the scope of this review.

In static systems the topology is typically expressed by the structure of the eigenvectors $|u_n(\mathbf {k})\rangle$ of the $n$th band of $H_{\mathrm {B}}$. In one dimension the often used-Zak phase [39] incorporates these Bloch states $|u_n(\mathbf {k})\rangle$ to express the topology of the system. The Zak phase is up to a factor of $2\pi$ a one-dimensional winding number

Such a winding number can be used in Floquet systems, too. However, the topology there is encoded in the propagators $U=\mathbb {P}\exp \left (\mathrm {i}\int H_{\mathrm {B}}(\tau )\mathrm {d}\tau \right )$, where $\mathbb {P}$ denotes the path ordering operator. The corresponding winding number for a one-dimensional setup yields to [40]

Localization and scatter-free transport at the boundary of topological systems are the striking phenomena of this field of physics. By considering an edge geometry the corresponding bulk Hamiltonian changes to an edge Hamiltonian $H_{\mathrm {E}}$, which normally is reduced by one degree of freedom in the momentum space [50,51]. If the two different systems at the boundary do not coincide in their topological invariants, states appear in the band gap of this edge structure. In real space these gap states are localized at the boundary/edge of the system [52]. The topological invariant of the bulk geometry $H_{\mathrm {B}}$ is related to the band gap states of $H_{\mathrm {E}}$ [53,54]. This relation is called bulk edge correspondence, since the invariant, a property of the bulk structure, predicts the states in the edge geometry [4,10]. In other words, the boundary truncates the topological structure of the bulk and this results in states which are not part of the bulk and therefore localized at the boundary. In two dimensions, these band gap states cross the entire band gap. Due to their involved topological origin, these states possess strictly monotonic slopes [55]. Hence, a reflection of these states in backward direction is not supported and they have to propagate in one predetermined direction. This results in a unidirectional transport behavior at the boundary of the system. Since the topological invariants do not change under the influence of slight perturbations, the bulk edge correspondence always maintains the unidirectional character of the edge state. It may be noted that this notion of a bulk edge correspondence is also valid for periodically driven systems. However, as discussed above for the topological invariants, one needs to take the whole Floquet period into account [41], since the band gaps might close. Otherwise one might encounter contradicting observations like the existence of edge states despite a vanishing invariant [41]. In essence, the topology of the bulk results in a robustness of the edge states. These edge states, especially the unidirectional transport, enable a plethora of applications in photonics [9].

## 3. Integrated-optical platforms

#### 3.1 Femtosecond laser-written waveguides

The predominant platform for topological photonics in integrated structures are coupled waveguide structures that were fabricated using the femtosecond laser direct-write (fsLDW) technology [56,57] (see Fig. 1(a)). When ultrashort laser pulses are tightly focused into a transparent bulk material, nonlinear absorption takes place leading to optical breakdown and the formation of a micro plasma, which induces a permanent change in the molecular structure. In certain materials such as fused silica or borosilicate glass, a local densification occurs, yielding to an increase of the refractive index [58]. By moving the sample transversely with respect to the focal spot, a waveguide is formed within the bulk material along the respective trajectory. Extended systems can be inscribed sequentially with this technology, providing the highest possible degree of flexibility in addressing the parameters of each individual waveguide. The hopping between the individual guides is adjusted by the refractive index of the guides as well as of their spacing, which can be precisely tuned. Therefore, a variety of waveguide arrangements can be realized in 1D and 2D, ranging from fully periodic homogeneous and inhomogeneous lattices [59,60] with hard edges [61–64] to the introduction of individual defects [65,66], disorder [67,68] and even waveguide crossings and junctions [69,70]. Moreover, the light evolution in the structure can be directly imaged using fluorescence microscopy [71,72]. As another benefit of fsLDW, the fabricated waveguide structures are stable and, hence, permanently inscribed into the bulk material. However, there are a few challenges that have to be overcome when working with femtosecond laser written waveguides. The first is that typical refractive index contrasts of the waveguides are on the order of $\Delta n \approx 10^{-3}$, which is comparably small to the surrounding medium (e.g. fused silica, $n_0\approx 1.45$). Hence, bending radii of the waveguides have to be chosen relatively large if radiative losses are to be avoided [73]. Moreover, fsLDW structures typically feature an extreme aspect ratio between their centimeter-scale length and the micron-scale width of the individual modification [74]. And, last but not least, the effective nonlinearity in such structures is rather small [75], such that intense femtosecond laser pulses are required in order to observe nonlinear propagation dynamics [76–78]. As a result, a delicate balance has to be struck between the required strength of evanescent coupling and reaching the desired nonlinear excitation regime, which may lay close to the damage threshold of the host material [79]. Regardless, waveguide arrangements fabricated with the fsLDW technology [80] constitute an exceptionally versatile testbed for the implementation and observation of topological phenomena in integrated photonic structures.

#### 3.2 Photolithographic waveguides

Another promising platform are lattices fabricated by lithographic techniques on semiconductor-on-insulator. On a semiconductor substrate a light confinement region is sandwiched between cladding materials. By etching the upper cladding, the refractive index change induces a planar evanescently coupled waveguide lattice [81]. This method can be improved by designing coupled ring resonators instead of waveguides (see Fig. 1(b)), commonly called CROWs (coupled resonator optical waveguides) [82,83]. The main benefit of these structures is the extraordinary high refractive index contrast of the waveguides, allowing for minuscule radii. At the same time, the Q factor of such resonators routinely exceeds $10^{4}$, such that complex geometries are possible while maintaining an extremely compact footprint in the sub-millimeter range. In this vein, the implementation of the Harper-Hofstadter model becomes feasible [84], in order to generate a non-trivial flux. The light distribution inside the structures as well as the transmission spectra can be imaged by conventional optical microscopy [82]. A major advantage of CROW structures is that they can be active [85] and strongly nonlinear [86]. However, two key challenges remain: For once, CROW structures are intrinsically planar. Moreover, CROWs emulate static systems with intact time-reversal symmetry. In particular this means that no Floquet modulation of the Hamiltonian for breaking time-reversal symmetry is possible, thus precluding a substantial subset of the topological physics of light from being implemented in such structures.

#### 3.3 Induced waveguides in photorefractives

The optical induction technique relies on the interference of several monochromatic light beams, whereby the resulting intensity pattern is translated into a (periodic or quasiperiodic) change in the refractive index of a photosensitive nonlinear material, usually a photorefractive Strontium-Barium-Niobate (SBN) crystal (Fig. 1(c)). The waveguide lattice is induced by plane waves arranged around the photorefractive crystal such that they interfere to produce a commonly periodic intensity pattern [87,88]. By adding stochastic phases to the individual waves results in a disorder in the waveguide depth, which is ideal for observing stochastic phenomena such as Anderson localization [89]. A spatially incoherent beam serves as the background illumination in order to optimize the photorefractive effect. The index modulation depth (and, hence, the coupling between adjacent waveguides) is controlled by applying a bias electric field to the crystal. Typically this results in an index contrast of the waveguides in the range of $10^{-3}$. The advantages of this technology include the ability to simultaneously create extended lattices comprised of hundreds of waveguides. Moreover, the underlying photorefractive induction mechanism allows the systems to be dynamically reconfigured on short time scales [89]. The main advantage, however, is the large nonlinearity, which has been instrumental in the observation of a plethora of soliton phenomena [90,91]. Drawbacks include the intricate interference patterns required to implement sharp edges [92], and the limited size of available SBN: Typically, the propagation dynamics extends only over a few coupling lengths [93]. Finally, the ephemeral induced waveguide arrangements dissipate quickly after their underlying intensity pattern is switched off.

#### 3.4 Laser-written polymer waveguides

An emerging approach to realize complex waveguide geometries is the fabrication of microscopic photonic structures using laser-assisted two-photon polymerization technology (Fig. 1(d)). There exist two different methods. The first involves the creation of the inverse of the waveguide structure, which is inscribed into a negative-tone photo resist [94,95]. After development, the hollow structure is infiltrated with another polymer (such as SU8-2) in order to create the waveguides, which can be solidified using appropriate heating. The resulting refractive indices of the outside material and the waveguide core are typically $n_0 = 1.54$ and $n_{\textrm {core}} = 1.59$, respectively. The other approach simply employs different laser powers for the development of the waveguides are the surrounding material [96]. There are several advantages of this platform. First, the shape of the waveguides can be precisely tailored, such that their properties are well known (such as the guides modes). The samples are extraordinary small, in the few-micron regime. There is a large dynamic range of the refractive index different of the waveguides to the environment, from $\Delta n = 8\cdot 10^{-3}$ to $5\cdot 10^{-2}$. As high refractive index waveguides are possible, the radiation losses in curved structures are minimal. However, there are still some challenges. A main issue is that there is no fluorescence excited by the propagating light, such that the intensities can only be measured at the end facet of the structure. Moreover, if one uses the two-step approach involving the filling of the inverse waveguide structure, all waveguides exhibit the same refractive index; a tailored detuning is not possible.

#### 3.5 Surface plasmon polariton waveguides

Dielectric-loaded surface plasmon polariton waveguides (DLSPPWs) commonly consist of poly(methyl methacrylate) (PMMA) ridges that are deposited via negative-tone gray-scale electron beam lithography on top of a 50-nm-thin gold film with typical widths of $w\approx 300\,\mathrm {nm}$ and thicknesses $h\approx 150\,\mathrm {nm}$ (see Fig. 1(e)) [98,99]. Due to the gray-scale lithography process, the thickness h of the PMMA ridge can be used as a parameter to conveniently adjust the DLSPPW effective refractive index in the range between $n_{\mathrm {eff}}\approx 1.05$ and $n_{\mathrm {eff}}\approx 1.25$. Numerical calculations show that a single DLSPPW may indeed support only one single plasmonic mode, such that single-band dynamics is achieved and the coupling between adjacent DLSPPWs depends on the overlap of the evanescent fields of adjacent guides. The main benefit of such structure are that the they are extremely small, which is highly beneficial for integrating topological phenomena into photonic devices. Moreover, the light distribution inside the samples can be directly monitored using standard nano microscopy. The challenges for this technology are the comparable high losses due to the employment of surface plasmon polaritons. Moreover, being tied to the metal surface, the structures are intrinsically planar, such that only 1D phenomena can be explored.

## 4. One-dimensional settings

#### 4.1 Static systems

The majority of topologically non-trivial 1D systems realized with static waveguide arrays are based on the Su-Schrieffer-Heeger (SSH) model [100], since it possesses chiral symmetry and the time reversal operator squares to one [15]. The SSH model describes a bipartite lattice, with alternating next neighbour hopping. The topological phase of this model depends on the ratio of the two involved coupling strengths. Due to the bulk edge correspondence the topology of the bulk determines the presence or absence of edge state at the system’s boundary.

The first time that these edge states were experimentally realized in photonics was in the context of Shockley- and Tamm surface states, using a superlattice imprinted in a photorefractive crystal [101], since the topological edge states can also be considered as Shockley-like states (see Fig. 2). In the same context it was investigated how the topological protection is influenced by on-site potentials and nonlinear effects [93]. While the discussion is originally applied to the transition from Shockley-like to Tamm-like surface states, it actually corresponds to the transition from a topological non-trivial system protected by chiral symmetry to a topological trivial system, where chiral symmetry is broken.

In another *quasi*-1D arrangement of waveguides in a
rhomboid structure, the topology is introduced by implementing a flux
in each plaquette [102] by
inserting a negative coupling [103]. If the structure is designed such that exactly half a
flux quantum is realized in each plaquette, all energy bands collapse
into flat bands, and all eigenmodes are localized and compact. This
localization due to a particular magnetic flux is known as
Aharonov-Bohm caging, as it is reminiscent of the famous Aharonov-Bohm
effect of electrons [104].
Even though this optical arrangement differs strongly from the SSH
model, it is related to the same topology through a square root
operation, such that it forms a *square-root
topological insulator* (see Fig. 3). In a subsequent work the a similiar caging
effect was achieved by harnessing higher spatial modes in polymer
waveguides [96].

Yet another concept that utilizes the framework of symmetry transformations to connect different SSH-type lattices is used in [105], where discrete supersymmetric transformations [106,107] are used to trigger topological phase transitions as well as transformations connecting edge and interface states.

Besides SSH-type lattices, also quasi crystalline structures were investigated by implementing the Fibonacci chain and connecting it to the Harper model [108]. This seemingly unfitting arrangement clearly shows that the underlying aspects of topology are fundamental and does not depend on the exact model. If band gaps of the Fibonacci chain are connected with band gaps of the Harper model possessing the same topology [81], no interface states exist, whereas they emerge if the topology differs.

Conventional wisdom holds that topological transitions should be observed at the edges of a lattice due to the bulk-boundary correspondence. However, it is indeed possible to observe topological transitions also at the bulk. In [109] it was shown that the winding number of the bulk can be retrieved by only measuring observables of pure bulk dynamics and averaging them over the propagation distance. Importantly, as the the SSH model is often considered as an archetypal example for topological systems, it is often used to study the impact of non-Hermitian concepts on topology. When considering an SSH lattice with loss on every other site it was shown in [110] that in such a system it is quite straight forward to characterize the topology of the bulk without any knowledge about the edges and the states on them. A simple measurement of the average displacement of the wave function is the only requirement to compute the winding number of the system and, hence, to retrieve all its topological features.

A particular instance of non-Hermiticity is $\mathcal {PT}$-symmetry [112], which causes a purely real eigenvalue spectrum despite the fact that the lattice potential is complex. It was quite debated whether $\mathcal {PT}$-symmetric systems can exhibit topological insulator phases at all [113]. However, the discussion was ended when a $\mathcal {PT}$-symmetric topological edge state was experimentally demonstrated at the interface of two non-Hermitian SSH lattices [111] (see Fig. 4). Moreover, this work also revealed that the exact definition of topological invariants in the non-Hermitian case is quite intricate, as many assumption stemming from Hermiticity do not strictly hold.

#### 4.2 Driven systems

Although the natural implementation of topological effects in 1D takes place in static SSH-based models, there are several incarnations of driven topological systems with only one spatial dimension. A standard example in condensed matter physics for a driven topological 1D system is the Thouless pump, a simple but powerful concept in topology [36]. This model considers the transport of charges due to adiabatic moving of a crystalline potential. Once can show that the pumped charge is quantized and, hence, topological. This concept can be readily adapted to photonics, where Thouless pumping was realized in a quasi-periodic Harper model (see Fig. 5) [81] and a Fibonacci chain [114]. To this end, the topological edge states on one side of the lattice were pumped towards the other by appropriately tuning the waveguide separation along the sample and using a simple single site excitation. Notably, despite being 1D in its implementation, the edge states in the Harper model and the Fibonacci chain are described by a Chern number, which requires at least two dimensions. The solution to this apparent contradiction is that both models can in fact be mapped onto the lattice version of the 2D integer quantum Hall effect [81]. In this vein, Thouless pumping of the edge states through the 1D system is, therefore, actually a process in 2D.

A 1D version of the well-known Floquet topological insulator was demonstrated in [115]. There, the driving is achieved by longitudinally changing the coupling between adjacent lattice sites, such that the waveguides are alternately coupled to the left and the right neighbor (see Fig. 6(a)), similar to an optical ratchet [116]. Due to this design, the system exhibits a periodicity in both the transverse and the longitudinal direction (i.e. along the waveguides), and a full Bloch-Floquet analysis can be applied. The appropriate invariant describing the system is the winding number, characterizing the emerging edge states as topological. It turns out that, for different coupling strengths between adjacent lattice sites, a phase diagram can be plotted, showing distinct regions of topologically trivial and non-trivial regimes in the system. Moreover, the ratchet system also exhibits drifting bulk states, which are another photonic manifestation of Thouless pumping (see Fig. 6(b-d)).

The experimental realization of a uniform synthetic magnetic flux in rhombic plaquettes due to periodic driving is reported in [117]. By combining a linear detuning along the lattice with a periodic modulation of the detuning along the propagation direction, a tunable effective nonzero magnetic flux in each plaquette is achieved. The two aspects of this engineered modulation-assisted tunnelling process are experimentally realized by circularly curving the lattice, and by periodically modulating the refractive indices along the propagation direction, respectively. A parabolic bending of the structure in the longitudinal direction together with a modulation of the refractive indices causes an engineered modulation-assisted tunnelling process that produces a tunable effective nonzero magnetic flux in each plaquette. If the structure is designed such that exactly half a flux quantum is realized in each plaquette, a similar effect as in [102] takes place and all eigenmodes are localized and compact. Interestingly, in the photonic setting of [117] the emerging edge states can be continuously connected to the topological edge states of the Creutz ladder [118], which sets a fascinating connection between AB cages in a lattice and the famous bulk-edge correspondence.

Commonly, robustness of topological protection is discussed in the context of static perturbations. In [98], the question of how dynamic perturbations impact the stability of topologically protected edge states was addressed. To this end, a particular setting was designed with two SSH lattices attached due to an interface waveguide. By modulating this waveguides either in the vertical plane (which keeps the coupling to both sites of the interface symmetric) or the horizontal plane (which causes an out-of-phase modulation of the coupling) results in three regimes of stability. For small perturbation frequencies, the system can follow adiabatically, and the topological edge state remains stable. Also for very high frequencies, the edge state remains stable, as the modulation essentially creates an effective detuned defect. For intermediate frequencies, however, the population of the edge state rapidly escapes into the bulk. This can be intuitively understood by arguments emerging from Fermi’s Golden Rule, but also from a more rigorous discussion incorporating coupling of Floquet replicas of the edge state eigenvalue to the bulk bands. This provides an interesting approach to control topology-mediated localization and the steering of light via an external parameter.

## 5. Two-dimensional settings

#### 5.1 Static systems

In 2011 an elegant concept was proposed to design two-dimensional arrangements of CROWs so as to support characteristic dynamics associated with quantum spin Hall Hamiltonians, namely the Hofstadter butterfly spectrum and robust edge state transport [84]. In their square lattice of coupled “site” resonators, asymmetric connecting waveguide loops (“link” resonators) serve to induce directionality in their system despite in the absence of external magnetic fields or time reversal symmetry breaking. As illustrated in Fig. 7(a), the path differences between the upper and lower branch of the link resonators imbue each plaquette, comprised of four site resonators at its corners, with the desired phase. Moreover, in this type of configuration, a judicious placement of scatters in the site- or link-resonators can mimic in-plane magnetic fields and spin-hopping terms, respectively. In 2013, the proposed CROW lattices were implemented on a silicon photonic platform (see Fig. 7(b)), marking the first observation of topological edge states of light in a 2D system [82] (see Fig. 7(c)).

Earlier that year, an entirely different approach to induce pseudomagnetic fields in photonic structures had been employed [119]. Adapting the strain engineering technique first proposed in [120], an inhomogeneous deformation to honeycomb-lattices of evanescently coupled waveguides, so-called “photonic graphene” (see Fig. 8), was introduced. In a fsLDW-based setting, the optical incarnation of Landau levels and the corresponding emergence of band gaps as well as a strong suppression of transverse transport at conditions equivalent to a magnetic flux of $5500\,\mathrm {T}$ were observed. Shortly thereafter, photonic graphene was employed to demonstrate how unidirectional strain can bring pairs of Dirac points in the band structure into close proximity to one another, and to eventually force them to merge. This transition coincides with annihilation of the edge states at the “bearded” edge, while simultaneously giving rise new states along the previously vacant “zigzag” edge [121]. A closer experimental examination of the edge dynamics in photonic graphene led to the discovery of previously unknown, new type of Tamm-like edge state residing at the “bearded” edge near the Van Hove singularity [122]. The impact of structural disorder on the edge states of photonic graphene was subsequently investigated by [123], who employed the degrees of freedom provided by the fsLDW technique to show that structural disorder, that is deviations in the off-diagonal elements of the Hamiltonian, does in fact preserve the confinement of the zero-energy edge state. In contrast, composite disorder was shown to break the chiral symmetry and thereby cancels the protection afforded by it.

Furthermore, a third option exists to imprint phases on lattice plaquettes in absence of external magnetic fields: The coupling coefficient in tight-binding lattices, typically assumed to be a positive real-valued quantity, may in fact be furnished with a negative sign as well [103]. This sign flip can be readily implemented for each individual link in an extended array by introducing an appropriately detuned ancillary waveguide between the lattice sites. Going beyond the initial demonstration of an Aharonov-Bohm interferometer in an isolated hexagonal ring [103], this very same technique has also been employed to investigating the properties of extended quasi-planar chains of Aharonov-Bohm cages [102].

Topological photonics was taken into the active domain, by developing the framework of topological insulator lasers [124], which was simultaneously put into practice by [85] in InGaAsP microring resonator arrays (see Fig. 9(a,b)). By selectively distributing gain along the edge of an otherwise lossy bulk lattice, robust singe-mode operation in the target mode is achieved, and an increased slope efficiency compared to the trivial regime was reported (see Fig. 9(d,e)). In contrast to the class of topological arrangements introduced in [84], the unidirectionality of the edge state is not mediated by systematically graded path lengths in the waveguides that mediate light exchange between the resonators. Instead, the placement of S-shaped internal elements within the resonators introduces chiral losses that serve to suppress lasing in the unwanted direction (see Fig. 9(e,f)).

Topological edge states mediated by the valley-Hall effect were observed in fsLDW honeycomb lattices with broken inversion symmetry [125]. In this type of system, known from solid-state 2D material such as hexagonal boron nitrite (HBN), an on-site detuning between the two sublattices yields opposite Berry curvatures in the two valleys of the band structure. The substantially larger detuning range available in photonic lattices provides access to the fully-gapped regime even for zigzag-type domain walls. A different degree of freedom, the couplings in and between the unit cells of the structure, can give rise to defect states at armchair-type edges and corners. As observed in [126], configurations where the coupling between unit cells exceeds the intra-cell coupling exhibit topological protection for a subset of corner whose energy falls in the middle of the energy gap. A different type of topological protected corner states was observed in so-called “breathing” Kagome-lattices [127], in which the coupling on the “up”/“down” triangular plaquettes of the lattice is different, while the entire system remains static in along the waveguide axis. In such systems, the protected states reside at corners where the outermost waveguide is only weakly coupled to the bulk of the lattice.

In the context of establishing an experimental platform for the implementation of 2D non-Hermitian configurations based on fsLDW photonic lattices with locally induced losses, the interplay between non-Hermitian and topological phase transitions has been investigated. In addition to presenting the first realization of a two-dimensional $\mathcal {PT}$-symmetric crystal [128], it has been demonstrated how such discrete refractive index landscapes with alternating losses between their lattice sites support topological mid-gap states. Residing at the bearded edge, they can exist only in a certain subset of the $\mathcal {PT}$-broken domain of the phase space of strained complex photonic graphene.

#### 5.2 Driven systems

One important step towards photonic topological insulators was the investigation of photonic graphene under periodic modulations. Similarly to the selective cancellation of coupling directions [73] and the full suppression of transverse transport [129] in the case of 2D dynamic localization, it was found that partial as well as complete band collapse can be induced by appropriately designed periodically undulating or spiral waveguide trajectories [130].

As it turned out, the potential of helical modulations extends far beyond diffraction management: Its inherently chiral nature does in fact lend itself to generate topological behavior, due to a breaking of the time reversal symmetry. In this vein, the first photonic topological insulator free of external fields was proposed and experimentally demonstrated, by using helical modulations [131]. In this photonic graphene based system, it was possible to observed the characteristic unidirectional scatter-free transport of visible light along the edge of the lattice as well as around defects (see Fig. 10). In these Floquet topological insulators (FTI), the protection against disorder is mediated by the non-zero Chern number of the bands that is induced by the helical modulation. As such, it can readily be extended to shield the path entanglement of photons from the effect of lattice imperfections [132]. When the radius of the helical waveguide trajectories is increased, the effective amplitude of the driving field is increased. While at first the group velocity associated with the edge states grows in kind, even larger radii were observed to yield lower speeds, until eventually light is transported in the opposite direction [131]. This behavior has been invesigated and it was found that, as a Floquet system moves away from the weak-field regime, it undergoes a series of transitions through a succession of distinct topological phases [133].

Notably, FTI systems rely on a global modulation of the lattice. In the simplest case, each waveguide of the lattice follows the same helical trajectory around its average transverse position. So-called dynamic defects arise when individual lattice sites deviate from this pattern, be it by an absence of modulation, relative phase shifts of the spiral, or even an opposite helicity. Laser-written polymer waveguides have been employed to synthesize such dynamic defects and explore their influence on topological protection (see Fig. 11) [97]. These experiments showed that individual defects do not compromise the robustness of the chiral edge modes, as no coupling do bulk modes was observed even to the point where the defect waveguide is overlapping with its neighbors at times and light is scattered out of the waveguides.

Another topological phenomenon accessible in arrangements of helically modulated waveguides are so-called Weyl points, the band-crossings of hypothetical massless two-component relativistic fermions. As monopoles of the Berry curvature, these entities possess a topological charge. In their “type II” variety with strictly positive group velocity, Weyl points were observed at optical frequencies in 2017. The host system, a superposition of two square lattices with the same helicity but a phase shift of $\pi$ between their spiral trajectories, exhibits a characteristic conical bulk diffraction, while supporting surface states with Fermi arc-like dispersion [134]. In a recent follow-up work [135],losses were introduced to one of the sublattices by means of segmentation [136]. Similar to its effect on the Dirac points in $\mathcal {PT}$-symmetric photonic graphene [137], this non-Hermitian detuning serves to turn the Weyl points of the Hermitian lattice into exceptional rings and prevents the conical diffraction pattern from forming. It was also studied how the rich topological properties of type II Dirac systems can be harnessed to control the dynamics of localized edge states [138]. By employing graphene-like arrangements - a generalization of the graphene honeycomb lattice - it was shown how the effective interactions between the lattice vertices can be selectively addressed without distortions to the shape or geometry of the unit cell.

The unparalleled robustness of light transport against disorder in the underlying lattice is one of the most remarkable features of photonic topological insulators, and certainly is in large parts responsible for the intensity of research devoted to them in recent years. Nevertheless, even in these systems, disorder is generally regarded as nuisance and potential danger to their functionality: Sufficiently strong perturbations will eventually obliterate the bandgap hosting the desired topological edge states, and render the system topologically trivial as any transport within it becomes suppressed by Anderson localization. In contrast, so-called topological Anderson insulators [139] actually rely on a certain amount of disorder to drive them into their non-trivial phase. This seemingly paradoxical behavior was first observed by [140] in a photonic graphene based Floquet system where a detuning between the on-site potentials of the two sublattices is sufficient to keep a topological bandgap from opening in the homogeneous configuration. Only when substantial on-site disorder was introduced across the 399547, did its edge become conductive for light injected through an ancillary “straw”.

The possibility of introducing geometric phases plays a fundamental role for the observable topology. It could be confirmed in experiments, that even a non-Abelian phase can be observed in a photonic structure [141]. In this work a tripod structure with precisely controlled couplings was harnessed to reveal non-Abelian features in a degenerate dark space. Non-Abelian geometric phases have also been measured in extended structures, where a braiding of topological vortex modes in a displaced honeycomb lattice was measured [142].

The concept of a photonic topological pump previously demonstrated in one-dimensional chains of waveguides [81] was also extended to 2D arrangements [46]. In this work the judicious superposition of two such pump processes in orthogonal directions allows light propagating through the structure to sample over momenta in two additional synthetic dimensions to establish a genuine 2D topological pump capable of transferring photons between opposite edges or corners of the lattice. The resulting band structure is therefore characterized by “second Chern numbers” (topological invariants in 4D) that in turn enable a quantized Hall response in the bulk associated with 4D symmetry. From this perspective, the transfer of light conveyed by the pump regime can be interpreted as equivalent of charge pumping through the bulk of a 4D system, between two of its opposite 3D hyper-surfaces or two of its 2D hyper-edges, respectively.

A different approach to extending photonic topological insulators to higher dimensions was demonstrated by employing sinusoidal modulations as a resonant drive to implement coupling between the equally spaced supermodes of planar $J_x$ waveguide arrays [144]. This technique enabled them to realize the first photonic TI in virtual dimensions [143]. Assembled into a sequence with appropriate phase shifts between their individual undulations, a system of such driven $J_x$ planes allows light to circulate along the circumference of the lattice spanned between one spatial direction and the mode number. Along the spatial dimension, the population of the fundamental mode of the individual planes is driven in one direction, whereas the highest supermode moves in the opposite way. Similarly, “motion” along the mode number axis is mediated by the up-/down-conversion between modes at the opposite edges (see Fig. 12).

Periodically undulating or spiralling waveguide trajectories directly imprint virtual driving fields imprinted onto the lattice. However, topological properties associated with the Floquet regime can also be synthesized by periodically modulating the couplings within the lattice, as independently demonstrated in [145] and [146]. In these square lattices, the horizontal and vertical couplings are selectively enabled along the direction of propagation in a driving protocol that traps light in cyclotron-like orbits around individual plaquettes in the bulk, while giving rise to anomalous topological modes along the edge . In contrast to the high-frequency limit where light only experiences the averaged properties of the driven lattice, the topological properties of these systems can no longer be described by a Chern number, as this quantity turns out to be zero for all Floquet bands. Instead, the details of the micro-motion have to be taken into account by using a generalized winding number [41]. Since the Floquet regime requires a sequence of exact replicas of the temporal unit cell, it is possible to obtain the same dynamics by cycling light signals through one and the same unit cell, as was demonstrated in [147]. Placing a sample containing the fsLDW unit cell of the lattice in a resonator, an ultra-high-speed photon counting camera allowed them to directly capture snapshots of the wave packet evolution after each round trip through the system for up to 8 Floquet periods. When employing this state recycling technique, additional phase gradients representing constant external fields can be introduced by polishing the end facets in a specific angle with respect to the waveguides.

In electronic topological insulators, the protected edge states originate from time reversal symmetry (TRS) and occur in pairs of opposite chirality that can transport spin-polarized currents along the edge of the insulating bulk. The bosonic nature of photons therefore seems to preclude the existence of such counter-propagating edge states protected by TRS. While it is possible to superimposing two Chern-type bosonic topological insulators with opposite orientation into a $\mathbb {Z}+\mathbb {Z}$ system, any interaction between these sublattices would destroy the topological properties of the system and result in a conventional insulator. Recently, this limitation was circumvented by designing a Floquet driving protocol that incorporates effective fermionic TRS [148] by encoding the spin degrees of freedom into the lattice itself. The resulting system is characterized by a $\mathbb {Z}_2$ invariant, and exhibits an anomalous topological phase where both the Chern number as well as the Kane-Mele invariant vanish. Notably, instead of being fragile in the presence of interactions between the sublattices, the fermionic TRS that gives rise to the topological properties of this $\mathbb {Z}_2$ is inextricably linked to two such interaction steps in its driving protocol (steps 2 and 5 in Fig. 13(a)). As a result, both the perfect spin flip case and arbitrary spin rotations give rise to pairs of counter-propagating edge states, which can be selectively populated depending on the choice of the initially excited sublattice (see Fig. 13(b)).

## 6. Outlook: nonlinear and quantum systems

An exciting direction of integrated topological photonics is incorporating nonlinearity into the dynamics. The existence of self-localized states in the bulk of a topological honeycomb lattice was predicted and its features theoretically explored in [149]. Subsequently, the fundamental features of nonlinear edge waves were analyzed in [150], and the dynamics of strongly localized edge solitons, emerging from edge states in topological photonic waveguide lattices, was explored in [151]. A first pioneering work experimentally demonstrated the formation of solitons in an anomalous TI array [152], clearly showing the expected circular motions of the nonlinear states. This was followed by a work harnessing an effective index mismatch as additional degree of freedom in the anomalous TI driving protocol. The corresponding complex Floquet employs a judicious placement of negatively detuned directional couplers throughout the full driving cycle to synthesize a system that is topologically trivial under linear conditions, yet allows for high-power light beams to establish topological protection of unidirectional edge states by means of the Kerr nonlinearity (see Fig. 14) [153]. This first experimental demonstration of this paradigm goes beyond topological solitons, which bifurcate from linear topological states and therefore can only exist of the underlying structure is topological in the first place. Instead, the transient topological phase in this nonlinearity-induced TI, as it is locally brought about by the propagating high-intensity wave packet itself. In this vein, it becomes possible to switch topological protection on or of at will, simply by choosing an appropriate power level for each pulse.

Waveguides have proven to serve as versatile platform for the implementation of quantum gates, due to their flexibility of arrangements and trajectories, combined with a high degree of interferometric stability. Therefore, it seems natural to utilize the beneficial features of topology in the quantum realm. While theoretical works already explore how unidirectional topological edge transport can be applied to quantum states [132], the experimental works are up to now limited to one spatial dimension. In these studies the SSH model [154,155], as well as the Harper model [154–156] is implemented to investigate zero-dimensional edge states [156–158] and adiabatic pumping [158] with bi-photon states. Throughout all works it was confirmed that the topological features can be transferred to bi-photon quantum states and protecting quantum correlation and entanglement. With those works the foundation is laid for topological quantum effects, which go beyond the adaption of classical (single photon) effects.

On these grounds, an exciting perspective for integrated photonic topological devices is their use as a platform for novel approaches to store and process quantum information by exploiting topological effects. In the presence of nonlinear elements, light may form complex topologically protected states, on which quantum information could be encoded in order to perform quantum logical operations. With respect to conventional quantum information protocols, quantum computing based on topological states has the advantage that the protected states can neither be coupled nor mixed with each other by local disturbances. In this vein, using an all-optical platform will be extremely favorable in view of integration of the quantum processing unit into an optical communication network.

## Funding

Alfried Krupp von Bohlen und Halbach-Stiftung (Szameit); Deutsche Forschungsgemeinschaft (BL 574/13-1, SZ 276/19-1, SZ 276/20-1, SZ 276/9-2).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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