We propose a route towards creating a metamaterial that behaves as a photonic Chern insulator, through homogenization of an array of gyromagnetic cylinders. We show that such an array can exhibit non-trivial topological effects, including topologically non-trivial band gaps and one-way edge states, when it can be homogenized to an effective medium model that has the Berry curvature strongly peaked at the wavevector k=0. The non-trivial band topology depends only on the parameters of the cylinders and the cylinders' density, and can be realized in a wide variety of different lattices including periodic, quasi-periodic and random lattices. Our system provides a platform to explore the interplay between disorder and topology and also opens a route towards synthesis of topological meta-materials based on the self-assembly approach.
We extend results of [Liu et al. Proc. Roy. Soc. Lond. A 469, 20130240, 2013] on artificial magneto-electric coupling and magnetism in moderate contrast dielectric layered media via high-order homogenization to the three-dimensional setting. For this, we consider asymptotic expansions in the vector Maxwell system. We then illustrate effect of magneto-electric coupling with numerical simulations for a layered system alternating positively and negatively refracting index. We unveil hyperbolic behaviour and magneto-electric coupling and derive a sharper estimate for the anisotropic effective medi…
We present an accurate methodology for representing the physics of waves, for periodic structures, through effective properties for a replacement bulk medium: This is valid even for media with zero frequency stop-bands and where high frequency phenomena dominate. Since the work of Lord Rayleigh in 1892, low frequency (or quasi-static) behaviour has been neatly encapsulated in effective anisotropic media. However such classical homogenization theories break down in the high-frequency or stop band regime. Higher frequency phenomena are of significant importance in photonics (transverse magnetic …
In disordered two dimensional Chern insulators, a single bulk extended mode is predicted to exist per band, up to a critical disorder strength; all the other bulk modes are localized. This behavior contrasts strongly with topologically trivial two-dimensional phases, whose modes all become localized in the presence of disorder. Using a tight-binding model of a realistic photonic Chern insulator, we show that delocalized bulk eigenstates can be observed in an experimentally realistic setting. This requires the selective use of resonator losses to suppress topological edge states, and acquiring …
Topological physics in photonic systems have attracted great attentions in recent years. In this work, we theoretically study the one and two dimensional photonic quasicrystal resonator lattices characterized by the first and second Chern number, which show exotic boundary states within the photonic energy band gap. In particular, the second Chern number protected edge states has opened up new possibilities for realizing topological physics of dimensions higher than three in photonic systems, which is highly sought for. Such photonic systems can be easily experimentally realized in regular pho…
The Chern topological numbers of a material system are traditionally written in terms of the Berry curvature which depends explicitly on the material band structure and on the Bloch eigenwaves. Here, we demonstrate that it is possible to calculate the gap Chern numbers of a photonic platform without having any detailed knowledge of its band structure, relying simply on the system photonic Green function. It is shown that the gap Chern number is given by an integral of the photonic Green function along a line of the complex frequency plane parallel to the imaginary axis. Our theory applies to a…
Controlling light propagation using artificial photonic crystals and electromagnetic metamaterials is an important topic in the vibrant field of photonics. Notably, chiral edge states on the surface or at the interface of photonic Chern insulators can be used to make reflection-free waveguides. Here, by both theoretical analysis and electromagnetic simulations, we demonstrate that gyromagnetic hyperbolic metamaterials (GHM) are photonic Chern insulators with superior properties. As a novel mechanism, the simultaneous occurrence of the hyperbolic and gyromagnetic effects in these metamaterials …
We have characterized the robust propagation modes of electromagnetic waves in helical structures by the section Chern number that is defined for a two-dimensional (2D) section of the three- dimensional (3D) Brillouin zone. The Weyl point in the photonic bands is associated with a dis- continuous jump of the section Chern number. A spatially localized Gaussian basis set is used to calculate the section Chern numbers where we have implemented the divergence-free condition on each basis function in 3D. The validity of the bulk-edge correspondence in a 3D photonic crystal is discussed in relation…
The recent realizations of topological valley phase in photonic crystal, an analog of gapped valleytronic materials in electronic system, are limited to the valley Chern number of one. In this letter, we present a new type of valley phase that can have large valley Chern number of two or three. The valley phase transitions between the different valley Chern numbers (from one to three) are realized by changing the configuration of the unit cell. We demonstrate that these new topological phases can guide the wave propagation robustly along the domain wall of sharp bent. Our results are promising…
In 2D Chern insulators (2D CI), the topology of the bulk states is captured by a topological invariant, the Chern number. The scalar bulk-boundary correspondence (sBBC) relates the change in Chern number across an interface with the number of 1D chiral edge modes at the interface. However, 3D Chern insulators (3D CI) can be characterized by a Chern vector C = (Cx, Cy, Cz) and a more general vector bulk-boundary correspondence (vBBC) is needed to correctly predict the propagation of the surface modes. In this work the possible interfaces between 3D photonic CIs are explored, focusing on possibl…
Time Reversal Symmetry (TRS) broken topological phases provide gapless surface states protected by topology, regardless of additional internal symmetries, spin or valley degrees of freedom. Despite the numerous demonstrations of 2D topological phases, few examples of 3D topological systems with TRS breaking exist. In this article, we devise a general strategy to design 3D Chern insulating (3D CI) cubic photonic crystals in a weakly TRS broken environment with orientable and arbitrarly large Chern vectors. The designs display topologically protected chiral and unidirectional surface states with…
Despite great interest in the quantum anomalous Hall phase and its analogs, all experimental studies in electronic and bosonic systems have been limited to a Chern number of one. Here, we perform microwave transmission measurements in the bulk and at the edge of ferrimagnetic photonic crystals. Bandgaps with large Chern numbers of 2, 3, and 4 are present in the experimental results which show excellent agreement with theory. We measure the mode profiles and Fourier transform them to produce dispersion relations of the edge modes, whose number and direction match our Chern number calculations.
As an important figure of merit for characterizing the quantized collective behaviors of the wavefunction, Chern number is the topological invariant of quantum Hall insulators. Chern number also identifies the topological properties of the photonic topological insulators (PTIs), thus it is of crucial importance in PTI design. In this paper, we develop a first principle computatioal method for the Chern number of 2D gyrotropic photonic crystals (PCs), starting from the Maxwell's equations. Firstly, we solve the Hermitian generalized eigenvalue equation reformulated from the Maxwell's equations …
The classification of bandstructures by topological invariants provides a powerful tool for understanding phenomena such as the quantum Hall effect. This classification was originally developed in the context of electrons, but can also be applied to photonic crystals. In this paper we study the topological classification of the refractive index surfaces of two-dimensional photonic crystals. We consider crystals formed from birefringent materials, in which the constitutive relation provides an optical spin-orbit coupling. We show that this coupling, in conjunction with optical activity, can lea…
We study the polaritonic bandstructure of two-dimensional atomic lattices coupled to a single excitation of a surface plasmon polariton mode. We show the possibility of realizing topological gaps with different Chern numbers by having resonant atomic transitions to excited states with different angular momentum. We employ a computational method based on the recently proposed Dirichlet-to-Neumann (DtN) map technique which accurately models non-Markovian dynamics as well as interactions involving higher-order electric and magnetic multipole transitions. We design topologically robust edge states…
We proposed a group-theory method to calculate topological invariant in bi-isotropic photonic crystals invariant under crystallographic point group symmetries. Spin Chern number has been evaluated by the eigenvalues of rotation operators at high symmetry k-points after the pseudo-spin polarized fields are retrieved. Topological characters of photonic edge states and photonic band gaps can be well predicted by total spin Chern number. Nontrivial phase transition is found in large magnetoelectric coupling due to the jump of total spin Chern number. Light transport is also issued at the ε/μ misma…
Nonlinearities in lattices with topological band structures can induce topological interfaces in the bulk of structures and give rise to bulk solitons in the topological bandgaps. Here we study a photonic Chern insulator with saturable nonlinearity and show the existence of topological bulk solitons. The fundamental bulk solitons exhibit as semi-vortex solitons, where only one pseudospin component has a nonzero vorticity. The bulk solitons have equal angular momentum at different valleys. This phenomenon is a direct outcome of the topology of the linear host lattice and the angular momentum ca…
We present the fundamental model of a topological electromagnetic phase of matter: viscous Maxwell-Chern-Simons theory. Our model applies to a quantum Hall fluids with viscosity. We solve both continuum and lattice regularized systems to demonstrate that this is the minimal (exactly solvable) gauge theory with a nontrivial photonic Chern number ($C\neq 0$) for electromagnetic waves coupled to a quantum Hall fluid. The interplay of symmetry and topology is also captured by the spin-1 representations of a photonic skyrmion at high-symmetry points in the Brillouin zone. To rigorously analyze the …
The topology of typical Chern insulators is rooted in the periodicity of the system along two directions of real-space. In this article, we depart from this standard concept and demonstrate that a generic non-Hermitian photonic waveguide periodic along a single direction of real space can be regarded as a sub-component of an extended system with a synthetic dimension and with a nontrivial Chern topology. In particular, we show that the number of bands below a band-gap of a generic waveguide determines the gap Chern number of the extended system. It is theoretically and numerically demonstrated…
The bulk-edge correspondence links the Chern-topological numbers with the net number of unidirectional states supported at an interface of the relevant materials. This fundamental principle is perhaps the most consequential result of topological photonics, as it determines the precise physical manifestations of nontrivial topological features. Even though the bulk-edge correspondence has been extensively discussed and used in the literature, it seems that in the general photonic case with dispersive materials it has no solid mathematical foundation and is essentially a conjecture. Here, I pres…
Commonly, a two-dimensional topological insulator is characterized by non-zero Chern numbers associated with its band structure. In our work, we present the experimental demonstration of an anomalous topological insulator, for which the Chern numbers are always zero, and the winding number is the appropriate quantity describing the topological character of the system. We probe the robustness of the chiral edge states in such anomalous topological insulators and analyze the phase transition between topological and trivial phase.
In the recent years, there has been a drive towards the realization of topological phases beyond conventional electronic materials, including phases defined in more than three dimensions. We propose a way to realize 2nd Chern number topological phases with photonic crystals simply made up of defect resonators embedded within a regular lattice of resonators. In particular, through a novel quasiperiodic spatial modulations in the defect radii, a defect lattice possessing topologically nontrivial Chern bands with non-abelian berry curvature living in four-dimensional synthetic space is proposed. …
We calculate the static dielectric tensor of a periodic system of aligned anisotropic dielectric cylinders. Exact analytical formulas for the effective dielectric constants for the E- and H- eigenmodes are obtained for arbitrary 2D Bravais lattice and arbitrary cross-section of anisotropic cylinders. It is shown that depending on the symmetry of the unit cell photonic crystal of anisotropic cylinders behaves in the low-frequency limit like uniaxial or biaxial natural crystal. The developed theory of homogenization of anisotropic cylinders is applied for calculations of the dielectric propertie…
Photonic waveguide arrays provide an excellent platform for simulating conventional topological systems, and they can also be employed for the study of novel topological phases in photonics systems. However, a direct measurement of bulk topological invariants remains a great challenge. Here we study topological features of generalized commensurate Aubry-André-Harper (AAH) photonic waveguide arrays and construct a topological phase diagram by calculating all bulk Chern numbers, and then explore the bulk-edge correspondence by analyzing the topological edge states and their winding numbers. In c…
In recent years there has been a great interest in topological materials and in their fascinating properties. Topological band theory was initially developed for condensed matter systems, but it can be readily applied to arbitrary wave platforms with little modifications. Thus, the topological classification of optical systems is usually regarded as being mathematically equivalent to that of condensed matter systems. Surprisingly, here we find that both the particle-hole symmetry and the dispersive nature of nonreciprocal photonic materials may lead to situations where the usual topological me…
We study topological states of honeycomb photonic crystals in absence of inversion symmetry using plane wave expansion and finite element methods. The breaking of inversion symmetry in honeycomb lattice leads to contrasting topological valley indices, i.e., the valley-dependent Chern numbers in momentum space. We find that the topological corner states appear for 60$^\circ$ degree corners, but absent for other corners, which can be understood as the sign flip of valley Chern number at the corner. Our results provide an experimentally feasible platform for exploring valley-dependent higher-orde…
The field of topological photonics was initiated with the realization of a Chern insulator phase in a gyromagnetic photonic crystal (PhC) with broken time-reversal symmetry (T), hosting chiral edge states that are topologically protected propagating modes. Recent advances in higher-order band topology have discovered another type of topological state, as manifested by those modes localized at the corners of a sample, which are known as corner states. Here we report the realization of a quadrupole higher-order topological insulator phase in a gyromagnetic PhC, induced by a topological phase tra…
The Chern index characterizes the topological phases of nonreciprocal photonic systems. Unlike in electronic systems, the photonic Chern number has no clear physical meaning, except that it determines the net number of unidirectional edge states supported by an interface with a trivial mirror. Here, we fill in this gap by demonstrating that the photonic Chern number can be understood as a quantum of the light-angular momentum in a photonic insulator cavity. It is proven that for a large cavity, when the discrete spectrum can be approximated by a continuum, the spectral density of the thermal f…
Systems with strong light-matter interaction opens up new avenues for studying topological phases of matter. Examples include exciton-polaritons, mixed light-matter quasiparticles, where the topology of the polaritonic band structure arises from the collective coupling between matter wave and optical fields strongly confined in periodic dielectric structures. Distinct from light-matter interaction in a uniform environment, the spatially varying nature of the optical fields leads to a fundamental modification of the well-known optical selection rules, which were derived under the plane wave app…
Bounds on the Topological Charge of Photonic Systems
University of Lisbon – Instituto Superior Técnico and Instituto de Telecomunicações（里斯本大学 - 高等理工学院与电信研究所）；Instituto Universitário de Lisboa (ISCTE-IUL)（里斯本大学学院）；Centro de Física de Materiales, Universidad del País Vasco（巴斯克大学材料物理中心）
The Chern topological numbers of a material platform are usually written in terms of the Berry curvature, which depends on the normal modes of the system. Here, we use a gauge invariant Green's function method to determine from first principles the topological invariants of photonic crystals. The proposed formalism does not require the calculation of the photonic band-structure, and can be easily implemented using the operators obtained with a standard plane-wave expansion. Furthermore, it is shown that the theory can be readily applied to the classification of topological phases of non-Hermit…
The field of topological photonics emerged as one of the most promising areas for applications in transformative technologies: possible applications are in topological lasers or quantum optics interfaces. Nevertheless, efficient and simple methods for diagnosing the topology of optical systems remain elusive for an important part of the community. In this tutorial, we provide a summary of numerical methods to calculate topological invariants emerging from the propagation of light in photonic crystals. We first describe the fundamental properties of wave propagation in lattices with a space-dep…
Optical fiber is a ubiquitous and indispensable component in communications, sensing, biomedicine and many other lightwave technologies and applications. Here we propose topological one-way fibers to remove two fundamental mechanisms that limit fiber performance: scattering and reflection. We design three-dimensional~(3D) photonic crystal fibers, inside which photons propagate only in one direction, that are completely immune to Rayleigh and Mie scatterings and significantly suppress the nonlinear Brillouin and Raman scatterings. A one-way fiber is also free from Fresnel reflection, naturally …
The properties that quantify photonic topological insulators (PTIs), Berry phase, Berry connection, and Chern number, are typically obtained by making analogies between classical Maxwell's equations and the quantum mechanical Schrödinger equation, writing both in Hamiltonian form. However, the aforementioned quantities are not necessarily quantum in nature, and for photonic systems they can be explained using only classical concepts. Here we provide a derivation and description of PTI quantities using classical Maxwell's equations, we demonstrate how an electromagnetic mode can acquire Berry p…
The recent development of topological photonics has revealed a variety of intriguing phenomena such as Weyl degeneracy. However, topologically non-trivial degeneracy with higher dimension has not been reported in photonics. In this Letter, topologically charged photonic surface degeneracy in a helix structure possessing two-fold screw symmetry is demonstrated. Calculated Chern number and photonic Fermi arcs support topological phase of the surface degeneracy. We also show that the photonic system can work in both metamaterial and photonic crystal regime, which have an order of magnitude differ…
A recently-proposed class of photonic topological insulators is shown to map onto Chalker-Coddington-type networks, which were originally formulated to study disordered quantum Hall systems. Such network models are equivalent to the Floquet states of periodically-driven lattices. We show that they can exhibit topologically protected edge states even if all bands have zero Chern number, which is a characteristic property of Floquet bandstructures. These edge states can be counted by an adiabatic pumping invariant based on the winding number of the coefficient of reflection from one edge of the …
Achieving topologically-protected robust transport in optical systems has recently been of great interest. Most topological photonic structures can be understood by solving the eigenvalue problem of Maxwell's equations for a static linear system. Here, we extend topological phases into dynamically driven nonlinear systems and achieve a Floquet Chern insulator of light in nonlinear photonic crystals (PhCs). Specifically, we start by presenting the Floquet eigenvalue problem in driven two-dimensional PhCs and show it is necessarily non-Hermitian. We then define topological invariants associated …
We consider conditions under which photonic crystals (PCs) can be homogenized in the higher photonic bands and, in particular, near the $Γ$-point. By homogenization we mean introducing some effective local parameters $ε_{\rm eff}$ and $μ_{\rm eff}$ that describe reflection, refraction and propagation of electromagnetic waves in the PC adequately. The parameters $ε_{\rm eff}$ and $μ_{\rm eff}$ can be associated with a hypothetical homogeneous effective medium. In particular, if the PC is homogenizable, the dispersion relations and isofrequency lines in the effective medium and in the PC should …
We propose a method of measuring topological invariants of a photonic crystal through phase spectroscopy. We show how the Chern numbers can be deduced from the winding numbers of the reflection coefficient phase. An explicit proof of existence of edge states in system with nonzero reflection phase winding number is given. The method is illustrated for one- and two-dimensional photonic crystals of nontrivial topology.
Pseudo-spin and valley degrees of freedom (DOFs) engineered in photonic analogues of topological insulators (TI) provide potential approaches to optical encoding and robust signal transport. Here we observe a ballistic edge state whose spin-valley indices are locked to the direction of propagation along the interface between a valley photonic crystal and a metacrystal emulating the quantum spin Hall effect. We demonstrate the inhibition of inter-valley scattering at a Y-junction formed at the interfaces between photonic TIs carrying different spin-valley Chern numbers. These results open up th…
Here, we develop a gauge-independent Green function approach to characterize the Chern invariants of generic non-Hermitian systems. It is shown that analogous to the Hermitian case, the Chern number can be expressed as an integral of the system Green function over a line parallel to the imaginary-frequency axis. The approach introduces in a natural way the "band-gaps" of non-Hermitian systems as the strips of the complex-frequency plane wherein the system Green function is analytical. We apply the developed theory to nonreciprocal electromagnetic continua, showing that the topological properti…
We designed and fabricated a time-reversal invariant Weyl photonic crystal that possesses single Weyl nodes with topological charge of 1 and double Weyl nodes with a higher topological charge of 2. Using numerical simulations and microwave experiment, nontrivial band gaps with nonzero Chern numbers for a fixed kz was demonstrated. The robustness of the surface state between the Weyl photonic crystal and PEC against kz-conserving scattering was experimentally observed.
In this paper, we study valley degree of freedom in all dielectric silicon photonic graphene. Photonic band gap opening physics under inversion symmetry breaking is revisited by the viewpoint of nonzero valley Chern number. Bulk valley modes with opposite orbital angular momentum are unveiled by inspecting time-varying electric fields. Topological transition is well illustrated through photonic Dirac Hamiltonian. Valley dependent edge states and the associated valley-protected backscattering suppression around Z-shape bend waveguide have been demonstrated.
To a significant extent, the rich physical properties of photonic crystals are determined by the underlying geometry, in which the composed symmetry operators and their combinations contribute to the unique topological invariant to characterize the topological phases. Particularly, the inter- and intra-coupling modulation in the two-dimensional (2D) Su-Schrieffer-Heeger model yields the topological phase transition, and exhibit first-order edge localized states and second-order corner localized corner states. In this work, we use the geometric anisotropy into the 2D square lattice composed of …
