Semi-infinite Photonic Crystal Research Articles

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.

Light localization in one-dimensional photonic crystal heterostructures

Effects of both metamaterial inclusion and lack of inversion symmetry on the localized modes in one-dimensional photonic heterostructures under normal incidence are investigated. Each structure consists of two semi-infinite photonic crystals separated by an interface or by several layers. A general condition for light localization between semi-infinite crystals is given by parallelism between two-dimensional vectors in the electromagnetic (H, E) space. The case of a defective photonic crystal is recovered by joining crystals having the same unit cell. The defect modes display leftward and rightward damped oscillations with the same ratio, even when the unit cell lacks inversion symmetry. Their frequencies are found in good agreement with peaks of the transmittance through truncated structures. In heterostructures including metamaterials, an anomalous exponential decay of the magnetic field is found near one edge of the zero-〈n〉 gap.

Interface states and bound states in the continuum in photonic crystals with different lattice constants.

The existence of interface states at the boundary of two semi-infinite photonic crystals (PhCs) with different lattice constants are investigated systematically. Compared to the interface states in the two PhCs with the same period, a band folding effect is observed for the interface states inside the common band gap of the two PhCs with different lattice constants. We demonstrate that these interface states can be predicted by the surface impedance of the two PhCs. The dispersion of interface states can be determined by the condition of impedance matching combined with the band folding effect. Moreover, some part of the folded interface states penetrates the region of projected bulk bands, and they usually leak to the bulk and form resonant states. However, the interface state at the Γ point can be perfectly localized and becomes a bound state in the continuum (BIC) due to the symmetry mismatch. These findings may provide a general scheme for designing BICs in the PhC structures based on the interface states.

Exceptional points and their coalescence ofPT-symmetric interface states in photonic crystals

The existence of surface electromagnetic waves in the dielectric-metal interface is due to the sign change of real parts of permittivity across the interface. In this work, we demonstrate that the interface constructed by two semi-infinite photonic crystals with different signs of the imaginary parts of permittivity also supports surface electromagnetic eigenmodes with real eigenfrequencies, protected by $ PT $ symmetry of such loss-gain interface. Using a multiple scattering method and full wave numerical methods, we show that the dispersion of such interface states exhibit unusual features such as zig-zag trajectories or closed-loops. To quantify the dispersion, we establish a non-Hermitian Hamiltonian model that can account for the zig-zag and closed-loop behaviour for arbitrary Bloch momentums. The properties of the interface states near the Brillouin zone center can also be explained within the framework of effective medium theory. It is shown that turning points of the dispersion are exceptional points (EPs), which are characteristic features of non-Hermitian systems. When the permittivity of photonic crystal changes, these EPs can coalesce into higher order EPs or anisotropic EPs. These interface modes hence exhibit and exemplify many complex phenomena related to exceptional point physics.

Reflectivity of 1D photonic crystals: A comparison of computational schemes with experimental results

We report the reflectivity of one-dimensional finite and semi-infinite photonic crystals, computed through the coupling to Bloch modes (BM) and through a transfer matrix method (TMM), and their comparison to the experimental spectral line shapes of porous silicon (PS) multilayer structures. Both methods reproduce a forbidden photonic bandgap (PBG), but slowly-converging oscillations are observed in the TMM as the number of layers increases to infinity, while a smooth converged behavior is presented with BM. The experimental reflectivity spectra is in good agreement with the TMM results for multilayer structures with a small number of periods. However, for structures with large amount of periods, the measured spectral line shapes exhibit better agreement with the smooth behavior predicted by BM.

Dirac-like cones at k=0

Dirac cones and Dirac points are found at the K (K') points in the Brillouin zones of electronic and classical waves systems with hexagonal or triangular lattices. Accompanying the conical dispersions, there are many intriguing phenomena including quantum Hall effect, Zitterbewegung and Klein tunneling. Such Dirac cones at the Brillouin zone boundary are the consequences of the lattice symmetry and time reversal symmetry. Conical dispersions are difficult to form in the zone center because of time reversal symmetry, which generally requires the band dispersions to be quadratic at k=0. However, the conical dispersions with a triply degenerate state at k=0 can be realized in two dimensional (2D) photonic crystal (PC) using accidental degeneracy. The triply degenerate state consists of two linear bands that generate Dirac cones and an additional flat band intersecting at the Dirac point. If the triply degenerate state is derived from the monopolar and dipolar excitations, effective medium theory can relate this 2D PC to a double zero-refractive-index material with effective permittivity and permeability equal to zero simultaneously. There is hence a subtle relationship between two seemingly unrelated concepts: Dirac-like cone and zero-refractive index. The all-dielectric double zero-refractive-index material has advantage over metallic zero-index metamaterials which are usually poorly impedance matched to the background and are lossy in high frequencies. The Dirac-like cone zero-index materials have impedances that can tune to match the background material and the loss is small as the system has an all-dielectric construction, enabling the possibility of realizing zero refractive index in optical frequencies. The realization of Dirac-like cones at k=0 can be extended from the electromagnetic wave system to acoustic and elastic wave systems and effective medium theory can also be applied to relate these systems to zero-index materials. The concept of Dirac/Dirac-like cone is intrinsically 2D. However, using accidental degeneracy and special symmetries, the concept of Dirac-like point can be extended from two to three dimensions in electromagnetic and acoustic waves. Effective medium theory is also applicable to these systems, and these systems can be related to isotropic media with effectively zero refractive indices. One interesting implication of Dirac-like cones in 2D PC is the existence of robust interface states. The existence of interface states is not a trivial problem and there is usually no assurance that localized state can be found at the boundary of photonic or phononic crystal. In order to create an interface state, one usually needs to decorate the interface with strong perturbations. Recently, it is found that interface state can always be found at the boundary separating two semi-infinite PCs which have their system parameters slightly perturbed from the Dirac-like cone formation condition. The assured existence of interface states in such a system can be explained by the sign of the surface impedance of the PCs on either side of the boundary which can be derived using a layer-by-layer multiple scattering theory. In a deeper level, the existence of the interface state can be accounted for by the geometric properties of the bulk band. It turns out that the geometric phases of the bulk band determine the surface impedance within the frequency range of the band gap. The geometric property of the momentum space can hence be used to explain the existence of interface states in real space through a bulk-interface correspondence.

Tamm states at the interface between a conventional material and a one dimensional photonic crystal with metamaterials

Using the Green׳s function formalism, Tamm states of localized modes are investigated at the interface separating auniform conventional material and a one-dimensional semi-infinite photonic crystal consisting of a series of alternating conventional materials and metamaterials. We investigate how the presence of such metamaterials influences the band structure of collective modes that appear in the photonic crystal, with special attention to the power spectrum of collective excitations and the dispersion relations. It is shown that there is one localized backward TE mode with frequencies below the resonance frequency of the metamaterial magnetic permeability and above such frequency there are one forward TM, and two backward TM and TE localized modes.

Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems

Surface impedance is an important concept in classical wave systems such as photonic crystals (PCs). For example, the condition of an interface state formation in the interfacial region of two different one-dimensional PCs is simply Z_SL +Z_SR=0, where Z_SL (Z_SR)is the surface impedance of the semi-infinite PC on the left- (right-) hand side of the interface. Here, we also show a rigorous relation between the surface impedance of a one-dimensional PC and its bulk properties through the geometrical (Zak) phases of the bulk bands, which can be used to determine the existence or non-existence of interface states at the interface of the two PCs in a particular band gap. Our results hold for any PCs with inversion symmetry, independent of the frequency of the gap and the symmetry point where the gap lies in the Brillouin Zone. Our results provide new insights on the relationship between surface scattering properties, the bulk band properties and the formation of interface states, which in turn can enable the design of systems with interface states in a rational manner.

Tamm state of semi-infinite photonic crystal based on surface defect cavity with porous silicon and its refractive index sensing mechanism

A refractive index sensing structure based on the Tamm state of photonic crystal with surface defect is proposed by combing the Tamm state of semi-infinite photonic crystal with the optical sensing mechanism of porous silicon, in which the efficient bearing mechanism of the porous silicon is introduced into the surface defect cavity. The existence of Tamm state is demonstrated at the edge between the defect cavity and the periodical photonic crystal structure, and the total reflection in the defect cavity is formed by adjusting the incident angle. The resonant defect peak is obtained in the reflection spectrum by adding an absorbing medium into the defect cavity in order to reduce the reflectivity of the resonant wavelength. The full width at half maximum and the quality factor (Q value) can be optimized by adjusting the parameters of photonic crystal. Based on those results, according to the relationship between Goos-Hänchen phase shift and the resonant wavelength, the model for the relationship between the resonant wavelength and the effective refractive index variation of porous silicon adsorbing layer caused by the change of the refractive index of the sample is established, and its refractive index sensing characteristics are analyzed. The numerical simulation results show that the Q value can attain to 1429 and the sensitivity is about 546.67 nm·RIU-1, which can demonstrate the effectiveness of the structure design and provide some theoretical references for designing the refractive index sensors with high Q values and sensitivities.

Enhancing the absorption of graphene in the terahertz range

We study graphene on a photonic crystal operating in the terahertz (THz) spectral range. We show that the absorption of graphene becomes a modulated function of frequency and can be enhanced by more than three times at specific frequency values, depending on the parameters of the system. The problem of a semi-infinite photonic crystal is also solved.

An a posteriori error estimator for $$hp$$ -adaptive continuous Galerkin methods for photonic crystal applications

In this paper we propose and analyse an error estimator suitable for \(hp\)-adaptive continuous finite element methods for computing the band structure and the isolated modes of 2D photonic crystal (PC) applications. The error estimator that we propose is based on the residual of the discrete problem and we show that it leads to very fast convergence in all considered examples when used with \(hp\)-adaptive refinement techniques. In order to show the flexibility and robustness of the error estimator we present an extensive collection of numerical experiments inspired by real applications. In particular we are going to consider PCs with point defects, PCs with line defects, bended waveguides and semi-infinite PCs.

Validity of the effective-medium approximation of photonic crystals

A definition of the effective permittivity and permeability is proposed for two-dimensional photonic crystals. Expression for these effective parameters is deduced from the combination of the dispersion law with the formula for the reflection coefficient of a semi-infinite photonic crystal in the single-mode approximation. From these expressions, it is shown that the effective permittivity and permeability take purely real values when the truncation plane of the photonic crystal contains an inversion center of the infinite crystal. In addition, these effective parameters are demonstrated to be continuous and bounded when the crystal mode turns from propagating to evanescent only if the eigenmode fields fulfill specific symmetry conditions. Finally, the validity of the single-mode approximation and the effective-medium model of photonic crystals is investigated. In particular, arguments suggesting that this model fails for bands exhibiting negative refraction are presented.

Interface guided modes in a photonic crystal heterostructure with a complete photonic band gap

In this paper, a new kind of photonic crystal heterostructure (PCH) with a complete photonic band gap, which is composed of two semi-infinite photonic crystals, consisting of a triangular array of circular air holes in homogeneous dielectric Si and a honeycomb array of circular dielectric Si rods in air, respectively, is proposed and demonstrated. The interface guided modes in the PCH are investigated by using the plane-wave expansion method. The influences of longitudinal sideslipping and transverse displacement of the lattices on interface guided modes are discussed. The simulation results show that the guided modes can exist in the heterostructure without any shift of the lattices and the number of guided modes strongly depends on the geometric parameters of the heterostructure. It is anticipated that the guided modes in this heterostructure can be achieved by adjusting the relative transverse or longitudinal displacement of the lattices.

Tunable optical anisotropy in three-dimensional photonic crystals

Artificial optical birefringence can be realized in three-dimensional photonic crystals with a uniaxial structural symmetry: e.g., woodpile photonic crystals with a tetragonal lattice structure in the long-wavelength limit. The ordinary and extraordinary indices of refraction are determined from calculation of the reflection coefficient for a plane wave incident on the surface of a semi-infinite photonic crystal at different angles. We find that the anisotropy can be widely tuned by simply changing the width and thickness of the dielectric rod. A large relative negative anisotropy over 33% is found. A transition from positive anisotropy to negative anisotropy can be readily achieved. At certain parameters, a structurally anisotropic nanostructure can behave like an optically isotropic medium. Our study opens a window to use artificial nanostructures to create an arbitrary optical anisotropy that is not possible in natural crystals.

Accurate Model of Electromagnetic Wave Propagation in Unidimensional Photonic Crystals with Defects

Complete vectorial studies of electromagnetic wave propagation in complex media are difficult and computationally intensive. A mathematical continuous model can simplify some of such difficulties. Using this technique, we have modeled the propagation of electromagnetic plane waves through a semiinfinite photonic crystal with defects. The model is exact in the sense that the only approximations are the numerical accurate solution of ordinary differential equations and the simulation of defects by a high frequency sinusoidal continuous variation of the main crystal period.

Semianalytic approach for analyzing coupling issues in photonic crystal structures

A semianalytic approach based on previously derived closed-form expressions for the transmission and reflection matrices between a dielectric waveguide and a semi-infinite photonic crystal (PhC) waveguide is proposed for analyzing coupling issues in PhC structures. The proposed approach is based on an eigenmode expansion technique and introduces several advantages with respect to other conventional numerical methods such as a shorter computation time and the possibility to calculate parameters, such as the reflection into PhC structures, difficult to obtain with others methods. Two different examples are analyzed and results compared to finite-difference time-domain simulations to prove the usefulness of the proposed approach: (i) An especially designed two-defects configuration placed within a PhC taper to improve the coupling efficiency and (ii) a coupled-cavity waveguide coupled to a single-line defect PhC waveguide.

Optical properties and photonic modes in patterned semiconductor systems

The optical response of a photonic crystal slab, where a dielectric grating is the elementary building block, is computed in the semiclassical framework, and compared with the dispersion curves of the bulk. The formalism adopted for the calculation employs the eigenmodes of a single layer as basis set for electromagnetic field expansion; it allows us to compute photonic crystal slab optical response as well as dispersion curves. Recently, the method was generalized to take into account spatial dispersion effects close to an exciton resonance of the system, and it was possible also to study the transition from dissipative to dispersive regime for a system under Bragg condition. The role of bulk and surface waves in the system is discussed as a function of the layer number N in the range N = 1 ÷ ∞. The “mirror effect”, observed in rectangular self-sustained gratings, due to the interplay among traveling, evanescent and guided waves, is recovered. This effect can be preserved also in the semi-infinite photonic crystal for selected values of the physical parameters. The role of light polarization in 1D and 2D systems is briefly discussed.

Light propagation in finite and infinite photonic crystals: The recursive Green’s function technique

We report a computational method based on the recursive Green's function technique for calculation of light propagation in photonic crystal structures. The advantage of this method in comparison to the conventional finite-difference time domain (FDTD) technique is that it computes Green's function of the photonic structure recursively by adding slice by slice on the basis of Dyson's equation. This eliminates the need for storage of the wave function in the whole structure, which obviously strongly relaxes the memory requirements and enhances the computational speed. The second advantage of this method is that it can easily account for the infinite extension of the structure both into air and into the space occupied by the photonic crystal by making use of the so-called ``surface Green's functions.'' This eliminates the spurious solutions (often present in the conventional FDTD methods) related to, e.g., waves reflected from the boundaries defining the computational domain. The developed method has been applied to study scattering and propagation of the electromagnetic waves in the photonic band-gap structures including cavities and waveguides. Particular attention has been paid to surface modes residing on a termination of a semi-infinite photonic crystal. We demonstrate that coupling of the surface states with incoming radiation may result in enhanced intensity of an electromagnetic field on the surface and very high $Q$ factor of the surface state. This effect can be employed as an operational principle for surface-mode lasers and sensors.

Electromagnetic modes in semi-infinite photonic crystals

In a real photonic crystal, there exist three modes: propagation mode, evanescence mode and surface mode. Using ideal modal, semi-infinite photonic crystals, we study their effects on the transmission spectrum of photonic crystals, respectively. Because there exists only one air–crystal interface in a semi-infinite photonic crystal, no multiple reflection occurs and no evanescent modes and backward-propagating modes exist in this structure. The effects of the evanescence modes are studied by comparing the transmission spectrum of a finite-thickness photonic crystal slab and that of a semi-infinite photonic crystal. In addition, the effects of the backward-propagating modes are investigated using a coated semi-infinite photonic crystal structure. Finally, we study the effects of the surface modes, and find that the transmission spectrum of a semi-infinite photonic crystal is strongly dependent on its termination.

Behavior of light at photonic crystal interfaces

Band structures and Bloch modes give a generalized description of light in infinite photonic crystals. We show that the band structure and Bloch modes also contain the information necessary to find the amplitude and phase of light reflected and transmitted from interfaces in systems made using finite and semi-infinite photonic crystals. We obtain the equivalent of the Fresnel coefficients for photonic crystals. We use these coefficients to derive the reflection of light from a photonic crystal of finite size and the resonant modes of photonic crystal cavities and line defects. Results are given for ideal two-dimensional crystals, as well as crystals etched in semiconductor slab waveguides.