International Journal of Modern Physics B

As exotic phenomena in optics, topological states in photonic crystals have drawn much attention due to their fundamental significance and great potential applications. Because of the broken time-reversal symmetry under the influence of an external magnetic field, the photonic crystals composed of magneto-optical materials will lead to the degeneracy lifting and show particular topological characters of energy bands. The upper and lower bulk bands have nonzero integer topological numbers. The gapless edge states can be realized to connect two bulk states. This topological photonic states originated from the topological property can be analogous to the integer quantum Hall effect in an electronic system. The gapless edge state only possesses a single sign of gradient in the whole Brillouin zone, and thus the group velocity is only in one direction leading to the one-way energy flow, which is robust to disorder and impurity due to the nontrivial topological nature of the corresponding electromagnetic states. Furthermore, this one-way edge state would cross the Brillouin center with nonzero group velocity, where the negative-zero-positive phase velocity can be used to realize some interesting phenomena such as tunneling and backward phase propagation. On the other hand, under the protection of time-reversal symmetry, a pair of gapless edge states can also be constructed by using magnetic–electric coupling meta-materials, exhibiting Fermion-like spin helix topological edge states, which can be regarded as an optical counterpart of topological insulator originating from the spin–orbit coupling. The aim of this article is to have a comprehensive review of recent research literatures published in this emerging field of photonic topological phenomena. Photonic topological states and their related phenomena are presented and analyzed, including the chiral edge states, polarization dependent transportation, unidirectional waveguide and nonreciprocal optical transmission, all of which might lead to novel applications such as one-way splitter, optical isolator and delay line. In addition, the possible prospect and development of related topics are also discussed.

In this article, we review the recent progress in the implementation of synthetic gauge fields for photons and the investigation of new photonic phenomena, such as non-equilibrium quantum Hall physics. In the first part, we discuss the implementation of magnetic-like Hamiltonians in coupled resonator systems and provide a pedagogical connection between the transfer matrix approach and the couple mode theory to evaluate the system Hamiltonian. In the second part, we discuss the investigation of nonequilibrium fractional quantum Hall physics in photonic systems. In particular, we show that driven strongly interacting photons exhibit interesting many-body behaviors which can be probed using the conventional optical measurement techniques.

We provide a pedagogical account of an early proposal realizing fractional quantum Hall effect (FQHE) using coupled quantum electrodynamics (QED) cavity arrays (CQCAs). We start with a brief introduction on the basics of quantum Hall effects and then review the early proposals in the simulation of spin-models and fractional quantum Hall (FQH) physics with photons in coupled atom-cavity arrays. We calculate the energy gap and the overlap between the ground state of the system and the corresponding Laughlin wavefunction to analyze the FQH physics arising in the system and discuss possibilities to reach the ground state using adiabatic methods used in Cavity QED.

We review how broken symmetries affect optical properties in photonic analog of graphene, namely, honeycomb-lattice photonic crystals (PhCs). The spatial symmetry of the honeycomb lattice yields Dirac spectra at Brillouin zone corners without fine tuning of physical parameters. In addition, the "Dirac-mass" gap can be introduced by breaking the time-reversal symmetry and/or the space-inversion symmetry. These two symmetries are closely related to the topology of radiation fields in momentum space, and are linked with nontrivial edge states if the system has edges. We show that an effective Hamiltonian for photon obtained with the aid of group theory predicts a modulation of chiral edge states that are hardly implemented in electronic graphene. Numerical simulation of the honeycomb-lattice PhCs of infinitely-long cylinders, confirms the prediction fairly well.

We review several two- and three-dimensional metamaterial designs which serve as photonic simulators of topological phenomena in condensed matter physics. Based on a coupled-mode theory we show that lattices of interacting cavities can simulate Haldane lattices which exhibit one-way guided modes and mode coalescence in the absence of external magnetic field. Furthermore, periodic holey metals filled with a magnetoelectric material exhibit a nonreciprocal photonic band structure in the optical regime. Lastly, we show that a three-dimensional lattice of interacting cavities respecting time-reversal symmetry and a certain point-group symmetry can simulate a topological crystalline insulator.

We show that topological frequency band structures emerge in one-dimensional (1D) electromagnetic (EM) lattices of metamaterial components without the application of an external magnetic field. Such lattices can be cavity arrays coupled with metamaterial elements which generate alternate positive and negative-phase hopping strengths. The topological nature of the band structure manifests itself by the occurrence of Dirac points in the band structure, by the emergence of edge modes in finite arrays as well as by a nonreciprocal scattering spectrum. Specific EM designs for demonstrating the above phenomena are discussed.

A periodic lattice of optical ring resonators can act as a photonic topological insulator, with the role of spin played by the direction of propagation of light within each ring. Using finite-difference time-domain (FDTD) simulations, we compute the projected band diagram of the system and its transmission spectrum, and demonstrate the existence of the topological edge states. The FDTD results are in good agreement with the predictions of transfer matrix theory.

The condition for the formation of Dirac cones at arbitrary points in the Brillouin zone by the accidental degeneracy of two photonic bands was examined by the degenerate perturbation theory and group theory based on the spatial symmetry of two modes. The analysis was applied to a two-dimensional square-lattice photonic crystal with C_4v symmetry and the dispersion relation in the vicinity of the M point was examined. Exact agreement between the analytical and numerical calculation was obtained.

We present a multiple-scattering method in conjunction with supercell calculations to study the electromagnetic surface states in three-dimensional photonic crystal (PC) slab. Using our technique, we obtain the first prediction of Dirac-cone photonic surface state in some three-dimensional PC slabs. Such a state can be used to investigate some extremal transmission phenomena of electromagnetic waves near the Dirac point on the surface of the crystal, which is similar to the case of electron on the surface of topological insulators.

Gyromagnetic photonic crystal (GPC) offers a promising way to realize robust transport of electromagnetic waves against backscattering from various disorders, perturbations and obstacles due to existence of unique topological electromagnetic states. The dc magnetic field exerting upon the GPC brings about the time-reversal symmetry breaking, splits the band degeneracy and opens band gaps where the topological chiral edge states (CESs) arise. The band gap can originate either from long-range Bragg-scattering effect or from short-range localized magnetic surface plasmon resonance (MSP). These topological edge states can be explored to construct backscattering-immune one-way waveguide and other nonreciprocal electromagnetic devices. In this paper we review our recent theoretical and experimental studies of the unique electromagnetic properties of nonreciprocal devices built in GPCs. We will discuss various basic issues like experimental instrumental setup, sample preparations, numerical simulation methods, tunable properties against magnetic field, band degeneracy breaking and band gap opening and creation of topological CESs. We will investigate the unidirectional transport properties of one-way waveguide under the influence of waveguide geometries, interface morphologies, intruding obstacles, impedance mismatch, lattice disorders, and material dissipation loss. We will discuss the unique coupling properties between one-wave waveguide and resonant cavities and their application as novel one-way bandstop filter and one-way channel-drop filter. We will also compare the CESs created in the Bragg-scattering band gap and the MSP band gap under the influence of lattice disorders. These results can be helpful for designing and exploring novel nonreciprocal electromagnetic devices for optical integration and information processing.

We report a new one-way propagation effect of light which only occurs in multi-mode waveguide and can be controlled by adjusting the geometry of light source. Based on such effect, an integrated silicon waveguide composed of two planar waveguides with different thickness is proposed. The two-dimensional finite difference time domain (FDTD) simulation results show that complete signal isolation is achieved. Our results suggest that this integrated silicon waveguide may play an important role in future optical devices as long as the ideal source can be generated by the transformation optics method.

We present an extension of the layer-multiple-scattering method to photonic crystals of gyrotropic spheres in a homogeneous host medium. The efficiency of the method is demonstrated on specific examples of three-dimensional chiral structures and surfaces of crystals of plasma spheres in an external static uniform magnetic field that lack, simultaneously, time-reversal and space-inversion symmetries, and exhibit a nonreciprocal spectral response.