Two-dimensional Dielectric Photonic Crystals Research Articles

Wavelength division multiplexing based on the coupling effect of helical edge states in two-dimensional dielectric photonic crystals.

Photonic topological insulators with topologically protected edge states featuring one-way, robustness and backscattering-immunity possess extraordinary abilities to steer and manipulate light. In this work, we construct a topological heterostructure (TH) consisting of a domain of nontrivial pseudospin-type topological photonic crystals (PCs) sandwiched between two domains of trivial PCs based on two-dimensional all-dielectric core-shell PCs in triangle lattice. We consider three THs with different number of layers in the middle nontrivial domain (i.e., one-layer, two-layer, three-layer) and demonstrate that the projected band diagrams of the three THs host interesting topological waveguide states (TWSs) with properties of one-way, large-area, broad-bandwidth and robustness due to coupling effect of the helical edge states associated with the two domain-wall interfaces. Moreover, taking advantage of the tunable bandgap between the TWSs by the layer number of the middle domain due to the coupling effect, a topological Y-splitter with functionality of wavelength division multiplexing is explicitly demonstrated exploiting the unique feature of the dispersion curves of TWSs in the three THs. Our work not only offers a new method to realize pseudospin-polarized large-area TWSs with tunable mode-width, but also could provide new opportunities for practical applications in on-chip multifunctional (i.e., wavelength division multiplexing) photonic devices with topological protection and information processing with pseudospin-dependent transport.

Coexistence of quantum spin and valley hall effect in two-dimensional dielectric photonic crystals

The location and transmission of light is the core of modern photonic integrated device, and the proposal of topological photonics provides a new way of implementing optical manipulation. Topological photonic structures based on the quantum spin hall effect or quantum valley hall effect have the properties of immunity to defects and suppress backscattering, so they play a key role in designing novel low-loss photonic devices. In this work, we design a two-dimensional dielectric photonic crystal with time-reversal symmetry to achieve the coexistence of the quantum spin hall effect and the quantum valley hall effect in a photonic crystal. The design can be likened to an electronic system in which two pairs of Kramers simplex pairs are constructed to achieve a quadruple simplex pair in a photonic crystal. First, based on the method of shrinking and expanding the silicon pillars arranged in the honeycomb structure, the quadruple degeneracy point at the <i>Γ</i> point of the first Brillouin zone is opened, and the corresponding topologically trivial or non-trivial photonic band gap is formed,thereby realizing quantum spin hall effect. The expanded honeycomb lattice evolves into a Kagome structure, and then positive and negative perturbations are added to the Kagome lattice, breaking the spatial inversion symmetry of the Photonic crystal. When mirror symmetry is broken, different chiral photonic crystals can be created,leading the degeneracy point of the non-equivalent valleys <i>K</i> and <i>K'</i> in the Brillouin zone to be opened and a complete band gap to appear, thus realizing the Quantum valley hall effect. In the common band gap, topologically protected edge states are induced by nontrivial valley Chern number at the interface between two photonic crystals with opposite chirality. The numerical calculations show that unidirectional transport and bending-immune topological boundary states can be realized at the interface composed of topologically trivial (non-trivial) and positively (negatively) perturbed photonic crystals. Finally, a four-channel system based on the coexistence of the two effects is designed, The system is a novel electromagnetic wave router that can be selectively controlled by pseudospin degree of freedom or valley degree of freedom. This system provides a potential method for realizing the optical encoding and robust signal transmission, thereby providing greater flexibility for manipulating electromagnetic waves.

Pseudospin topological phase transition induced by rotation operation in two-dimensional dielectric photonic crystal with C6 symmetry

We construct a kind of two-dimensional photonic crystal with triangular lattice in which the primitive cell contains six identical rods composed of two half cylinders with different dielectric materials. All the rods have the same rotations around themselves, ensuring that the system preserves the C6 symmetry. The pseudospin topological phase transition (TPT) of the system is realized by rotating the rod elements, and the helical edge states (HESs) are constructed along the interface between two photonic crystals with opposite rotations. Numerical results and simulations reveal that the HESs are topologically protected with the pseudospin-locked unidirectional propagation of electromagnetic waves with robustness against corners and defects along the interface. Adjusting the size of rod elements, the TPT can be engineered at needed frequency.

Effective medium theory for photonic crystals with quadrupole resonances.

An effective medium theory is proposed to characterize two-dimensional dielectric photonic crystals (PCs) exhibiting quadrupole resonances. In addition to the effective permittivity and permeability associated with electric and magnetic dipoles, we obtain a local effective parameter to describe the contributions of quadrupole resonances by taking the low frequency limit of multiple-scattering theory. These effective parameters can be used to predict the characteristics of double-Dirac-cone PCs, showing good agreement with the numerical results. Moreover, we show that, after introducing the new effective parameter, the double-Dirac-cone PCs can be regarded as a generalization of the traditional double-zero-index metamaterials.

Dual index properties of photonic crystal and its application in subwavelength focusing

Based on the two-dimensional dielectric wedge-shaped photonic crystal, we studied the existence of two kinds of equivalent refractive indexes (negative refractive index is −1, positive refractive index is 1.897) in the photonic crystal by means of fast Fourier transform (FFT), as well as its influence on the optical propagation characteristics was discussed. By adjusting the wedge angle of the photonic crystal to eliminate the influence of positive refraction beam on the imaging, the sub wavelength focusing with half-width less than half of a wavelength can be realized. The numerical results show that the Full Width Half Maximum (FWHM) of the image point can reach a minimum of when the wedge angle of the photonic crystal is 73°.

Investigation of coupling effect between a unidirectional air waveguide and two cavities with one-way rotating state

In the paper we present an implementation of resonance coupling between a unidirectional air waveguide and two cavities with one-way rotating state in a two-dimensional dielectric photonic crystal and a magneto-optical photonic crystal. We theoretically calculate the dispersion curves of the line defect waveguide and the dispersion curves of waveguide-cavity interaction, respectively. Numerical simulations show that, via waveguide-cavity coupling, a series of interesting resonance phenomena can be generated in the system. By modifying direction of the applied external magnetic field, storing and discharging of photonic energy can be realized. Our work may provide a practical and useful guidance for optical communications.

Visualization of Higher-Order Topological Insulating Phases in Two-Dimensional Dielectric Photonic Crystals.

The studies of topological phases of matter have been developed from condensed matter physics to photonic systems, resulting in fascinating designs of robust photonic devices. Recently, higher-order topological insulators have been investigated as a novel topological phase of matter beyond the conventional bulk-boundary correspondence. Previous studies of higher-order topological insulators have been mainly focused on the topological multipole systems with negative coupling between lattice sites. Here we experimentally demonstrate that second-order topological insulating phases without negative coupling can be realized in two-dimensional dielectric photonic crystals. We visualize both one-dimensional topological edge states and zero-dimensional topological corner states by using the near-field scanning technique. Our findings open new research frontiers for photonic topological phases and provide a new mechanism for light manipulating in a hierarchical way.

Transmission properties in a photonic crystal with a Dirac-like point located at the Brillouin zone center

Light transmission properties of a two-dimensional dielectric photonic crystal (PC) with a Dirac-like point located at the Brillouin zone center are studied. The results show that at the Dirac-like point, the transmittance remains constant rather than inversely varying with the PC's thickness, that is, this transmission does not follow the pseudo-diffusive phenomenon. Moreover, since the PC with a Dirac point at the Brillouin zone center can mimic zero index material (ZIM), transmittance of a real homogeneous ZIM layer is also theoretically investigated. Invariant transmittance is obtained again and the constant transmittance at the Dirac-like point is confirmed. Physical mechanisms of the pseudo-diffusive transmission as well as the constant transmission are both analyzed. It is found that the Dirac point is a critical state between band gaps and transmission bands, where the wave vector takes a complex number. It is the imaginary part of the wave vector which causes the decrease of the transmittance. While at the Dirac-like point at the center of the Brillouin zone, an additional flat band makes the Dirac point frequency fall into a transmission band where the wave vector is a real number, therefore the transmittance does not display any decrease tendency.

Is the photonic crystal with a Dirac cone at its Г point a real zero-index material?

A novel optical property of a conventional two-dimensional dielectric photonic crystal (PC) has been reported by establishing a Dirac cone at the Г point of its band structure: it manipulates waves as if it had zero refractive indices at the Dirac point frequency (Huang et al. in Nature Mater 10:582–586, 2011). However, this PC is not a real homogeneous material. What is the difference between it and a real zero-index material (ZIM) is a fundamental issue of physics itself and also crucial to their applications. In this paper, through investigating the dispersion relations and the transmission properties of this kind of PC, we found that under normal incidence, at the frequency of the Dirac point, this PC does behave like a ZIM material. However, under oblique incidence, nonzero transmittances are obtained even at large incident angles, which is quite different from the real ZIM. The physical reason of the transmission properties was also investigated. It was found that near the Г point, there is an additional flat dispersion surface which intersects with the Dirac cone at the Dirac point. It is this flat dispersion surface which supports the Dirac point energy to exist in this PC, resulting in the nonzero transmission under oblique incidence.

On the mapping of Dirac-like cone dispersion in dielectric photonic crystals to an effective zero-index medium

Two-dimensional (2D) dielectric photonic crystals with accidentally induced Dirac-like points (DLPs) at the Brillouin zone center behave as an effective zero-index medium under certain conditions. We investigated different 2D square lattice dielectric PCs, such as dielectric rods of square and circular cross sections and core–shell rods in a square lattice, and cataloged the conditions under which these PCs can mimic an effective zero-index medium. We found that PCs with DLPs formed by dipole and quadrupole moments cannot be mapped to an effective zero-index metamaterial (ZIM) even if the DLP is at low frequency, indicating the applicability of effective medium theory. The effective ZIM can only be realized in PCs with DLPs if all three of the following conditions are fulfilled simultaneously: (i) the DLP is formed by monopole and dipole moments, (ii) the DLP is at low enough frequency, and (iii) there is only one propagating band in the extended states above the DLP frequency. The shape of the iso-frequency contours does not play a significant role toward the mapping of the DLP to an effective ZIM.

Geometric phase induced interface states in mutually inverted two-dimensional photonic crystals

The exposed surface of a truncated photonic crystal (PC) may or may not carry surface states. Likewise, when two PCs are joined together, there is no assurance that interface states can be found at the boundary. We show, however, that interface states must exist at the boundary between a two-dimensional dielectric PC and its ``inverted'' partner (the conjugate structure formed by interchanging the high/low dielectric region) as long as a common gap can be found above the lowest band and the crystal structures possess a mirror symmetry on average along the interface direction. The interface states form deterministically as a result of different geometric phases of the bulk bands across the boundary. As the existence is protected by topological principles, the interface state will persist even in the limit of extremely small common gaps. The presence of interface states is demonstrated theoretically for a variety of PCs and also experimentally.

Dirac-like cone dispersion in two-dimensional core-shell dielectric photonic crystals

The Dirac-like cone dispersion induced by accidental degeneracy is demonstrated in two-dimensional dielectric photonic crystals (PCs) of core-shell rods arranged in square and triangular lattices. The Dirac-like point (DLP) is achieved at the center of the Brillouin zone with a threefold degenerate state having two bands forming a Dirac cone and a third flat band intersecting the cone at the same frequency. This degenerate state is accidental and exists only for particular set of parameters of the PC. These parameters have been adjusted to obtain accidentally induced DLP. When the core region has a higher refractive index than the cladding, the DLP has monopole and dipole field configurations. When the core region has a lower refractive index than the cladding, the DLP is formed by quadrupole and dipole field configurations. The PCs exhibiting Dirac-like dispersion formed by monopole and dipole interactions can be mapped to an effective medium with ϵ and μ equal to zero. The isotropic behavior of DLP formed by monopole and dipole interactions is observed by computing isofrequency contours.

Double Dirac cones in two-dimensional dielectric photonic crystals.

By exploiting the accidental degeneracy of the doubly-degenerate dipolar and quadrupolar modes, we show that a two-dimensional dielectric photonic crystal (PC) can exhibit the double Dirac cone dispersion relation at the Γ point. Using a perturbation method and group theory, we demonstrate that the double cone is composed of two identical and overlapping Dirac cones with predictable linear slopes, and the linearity of the dispersion is guaranteed by the spatial symmetry of the Bloch eigenstates. Numerical simulations including wave-front shaping, unidirectional transmission and perfect tunneling show that the corresponding PC structure can be characterized by a zero effective refractive index.

Effect of longitudinal mode on the transmission properties near the Dirac-like point of the photonic crystals

Recently, Chan and his collaborators reported that a crossing point of bands can be achieved at the Brillouin zone center in two-dimensional (2D) dielectric photonic crystals (PhCs) by accidental degeneracy of modes. At the crossing point, the accidental threefold degeneracy of modes generates a Dirac cone and an additional flat band (longitudinal mode) intersecting the Dirac cone. This is different from that of the Dirac point at the corner of the hexagonal Brillouin zone in which only Dirac cone exists. As a result, the crossing point at the Brillouin zone center is called a Dirac-like point. If the accidental degeneracy occurs by a monopole mode and two dipolar modes, the dielectric PhCs can be mapped to a zero-refractive-index system in which the effective permittivity and permeability are zero at the Dirac-like point from the effective medium theory. According to the Maxwell equations, if the permittivity and permeability are zero, the optical longitudinal modes can exist, in additional to the well-known transverse modes. The additional flat band at the Dirac-like point is closely connected with the longitudinal mode. For a homogeneous zero-index material (ZIM), the flat band is dispersionless and the longitudinal mode cannot couple with the external light. But in a finite-sized PhC, there is always some spatial dispersion, so the flat band is not perfectly dispersionless when it is away from the zone center. Therefore, if the wave source is a Gaussian beam with non-zero k-parallel components, the longitudinal mode can be excited. And the effective wavelength of ZIM is extremely large, leading to many scattering properties. However, in a PhC which behaves as if it had a zero refractive index, it is very interesting to show how the longitudinal mode influences the wave propagations in the PhC when the longitudinal mode is excited. In this paper, the effect of longitudinal mode on the transmission properties near the Dirac-like point of PhCs is investigated by numerical simulation. The alumina dielectric rods can be moved randomly in the structure to result in the disorder of the structure. Our results show that the transmission properties at the Dirac-like point are very different from those near the Dirac-like point, when the longitudinal mode is excited. At the Dirac-like point, the transmittance decreases with increasing disorder, as a result of the influence of the longitudinal mode, which is similar to the one in the pass band. Above the Dirac-like point without the disturbance of longitudinal mode, the transmittance is insensitive to the disorder in the structure, so that the structure may mimic a near-zero index materials and have a large effective wavelength. These results may further improve the understanding about the optical longitudinal mode and the zero refractive material.

Transmission properties near Dirac-like point in two-dimensional dielectric photonic crystals

We study the wave transport properties near the Dirac-like point at the Brillouin zone center in two-dimensional dielectric photonic crystals with finite thickness. Both simulations and microwave experiments confirm that the transmittance is nearly inversely proportional to the length (L) of the samples in the propagation direction near the Dirac-like point. This transmittance law comes from the conically shaped dispersion. Since the conical singularity at the Brillouin zone center corresponds to zero refractive index, the field at the Dirac-like point contains a basic component of nearly uniform field. In contrast, the field at the Dirac point in the corner of the hexagonal Brillouin zone contains a basic component of inhomogeneous standing-wave–like field.

Antireflection structure for an effective refractive index near-zero medium in a two-dimensional photonic crystal

Two-dimensional dielectric photonic crystals (PCs) having periodic air hole cylinders, when designed properly, exhibit near-zero effective refractive index and the wave impedance is dependent on local observation points. The incident wave is mostly reflected at the PC-air interface due to large impedance mismatch. We show, analytically and numerically, that even in the near-zero effective refractive index case the reflection can be suppressed by utilizing an antireflection structure consisting of a PC with the same lattice constant but a different radius for the periodic air hole cylinders. The antireflection PC must be truncated at properly selected cross sections in order to possess the same impedance at cross sections with the host PC and with the air structure. An analytical model combined with the plane-wave expansion method captures the antireflection behavior obtained by the full wave simulations.

Epsilon-near-zero or mu-near-zero materials composed of dielectric photonic crystals

We theoretically demonstrate that at certain frequencies two-dimensional dielectric photonic crystals (PCs) may be regarded as either epsilon-near-zero or mu-near-zero materials. We show that the transmission through a slab of such materials upon normal incidence is normally non-unity and decays with slab thickness. However, when the incident angle increases slightly, the transmittance experiences a dramatic increase due to the Brewster effect. The combination of the tunneling and resonance effects makes such materials good candidates for almost perfect bending waveguides and cloaking in waveguides. The zero index also enables applications of focusing and directive emission. At last, the distinction between the single-zero and double-zero media is discussed. In all of the above results, the numerical simulations perfectly match with theoretical predictions from the effective medium analysis.

Selection rule for Dirac-like points in two-dimensional dielectric photonic crystals

We developed a selection rule for Dirac-like points in two-dimensional dielectric photonic crystals. The rule is derived from a perturbation theory and states that a non-zero, mode-coupling integral between the degenerate Bloch states guarantees a Dirac-like point, regardless of the type of the degeneracy. In fact, the selection rule can also be determined from the symmetry of the Bloch states even without computing the integral. Thus, the existence of Dirac-like points can be quickly and conclusively predicted for various photonic crystals independent of wave polarization, lattice structure, and composition.

Two-Dimensional Silicon Nitride Photonic Crystal Band Gap Characteristics

Plane-wave expansion method was employed to analyze the photonic band gap in two-dimensional silicon nitride photonic crystal. The effects of filling ratio and lattice structure type on the photonic band gap were studied. The results showed that two-dimensional dielectric cylinder type silicon nitride photonic crystal only has TE mode band gap, while, the air column type photonic crystal has complete band gap for TE and TM modes simultaneously. The distribution of band gap can be influenced by the filling ratio of dielectric materials and the lattice type. It is shown that the triangular lattice structure is much easier to form band gap than square lattice structure.

Spatial filtering using dielectric photonic crystals at beam-type excitation

Spatial filtering is demonstrated at beam-type excitations by utilizing finite thickness slabs of two-dimensional dielectric photonic crystals (PCs) showing exotic Fabry–Perot resonances that are preserved over a wide range of variation of the incidence angle. Bandstop and dual-bandpass filtering effects are illustrated theoretically and the corresponding filters are validated in the microwave experiments by using square-lattice PCs. It is shown that the basic transmission features that were observed earlier for a plane-wave illumination are also recognizable at beam-type excitations. The proposed spatial filtering mechanism exhibits directional beaming. The desired widths and the locations of the passbands and stopbands are attainable in the angle domain with a proper choice of the operating frequency for the given excitation characteristics.