Eigenvalue problems for electromagnetic resonant states on open dielectric structures are non-Hermitian and may have exceptional points (EPs) at which two or more eigenfrequencies and the corresponding eigenfunctions coalesce. EPs of resonant states for photonic structures give rise to a number of unusual wave phenomena and have potentially important applications. It is relatively easy to find a few EPs for a structure with parameters, but isolated EPs provide no information about their formation and variation in parameter space, and it is always difficult to ensure that all EPs in a domain of the parameter space are found. In this paper, we analyze EPs for a dielectric slab containing a periodic array of circular cylinders. By tuning the periodic structure towards a uniform slab and following the EPs continuously, we are able to obtain a precise condition about the limiting uniform slab, and thereby order and classify EPs as tracks with their endpoints determined analytically. It is found that along each track, a second order EP of resonant states (with a complex frequency) is transformed to a special kind of third order EP with a real frequency via a special fourth order EP. Our study provides a clear and complete picture for EPs in parameter space, and gives useful guidance to their practical applications.
A highly unusual pattern of strong multiple resonances for bulk electromagnetic waves is predicted and analysed numerically in thick periodic holographic gratings in a slab with the mean permittivity that is larger than that of the surrounding media. This pattern is shown to exist in the geometry of grazing-angle scattering (GAS), that is when the scattered wave (+1 diffracted order) in the slab propagates almost parallel to the slab (grating) boundaries. The predicted resonances are demonstrated to be unrelated to resonant generation of the conventional guided modes of the slab. Their physica…
We present an absorbing boundary condition for electromagnetic frequency domain simulations of photonic crystals and metamaterials. This boundary condition can simultaneously absorb multiple Bloch-Floquet eigenmodes of a periodic crystal, including both propagating and evanescent modes. The photonic crystal or metamaterial in question can include lossy, active, anisotropic and even bi-anisotropic inclusions. The absorbing boundary condition is dependent on an orthogonality condition for Bloch-Floquet eigenmodes, a generalized version of which is presented here. We test this absorbing boundary …
We study photonic band structure of two- and three-dimensional magnetophotonic crystals and the polarization properties of their eigenmodes using a plane wave expansion method. The alteration of the photonic band structure and eigenmodes by magnetization are examined. Two orientations are studied: in-plane magnetization and perpendicular magnetization. The magnetization-induced symmetry breaking effects are shown to be responsible for the formation of exotic magnetic Bloch modes, comprised of two or more coupled symmetric and antisymmetric Bloch modes of the corresponding non-magnetic photonic…
We show that, taking a two-dimensional photonic-crystal slab system as an example, surprisingly high quality factors (Q) over 10^5 are achievable, even in the absence of a rigorous photonic-band-gap. We find that the density of in-plane Bloch modes can be controlled by creating additional photon feedback from a finite-size photonic-crystal boundary that serves as a low-Q resonator. This mechanism enables significant reduction in the coupling strength between the bound state and the extended Bloch modes by more than a factor of 40.
The valleys of two-dimensional transition metal dichalcogenides (TMDCs) offer a new degree of freedom for information processing. To take advantage of this valley degree of freedom, on one hand, it is feasible to control valleys by utilizing different external stimuli like optical and electric fields. On the other hand, nanostructures are also used to separate the valleys by near field coupling. However, for both above methods, either required low-temperature environment or low degree of coherence properties limit their further applications. Here, we demonstrate all-dielectric photonic crystal…
A photonic-crystal slab can support bound states in the continuum (BICs) which have infinite lifetimes but embedded into the continuous spectrum of optical modes in free space. The formation of BICs requires a total internal reflection (TIR) condition at both interfaces between the slab and free space. Here, we show that the TIR of Bloch waves can be directly obtained based on the generalized Fresnel equations proposed. If each of these Bloch waves picks up a phase with integer multiples of 2pi for traveling a round trip, light can be perfectly guided in the slab, namely, forming a BIC. A BIC …
We present a systematic comparison between guided modes supported by slab waveguides and Bloch Surface Waves (BSWs) propagating at the surface of truncated periodic multilayers. We show that, contrary to common belief, the best surface field enhancement achievable for guided modes in a slab waveguide is comparable to that observed for BSWs. At the same time, we demonstrate that, if one is interested in maximizing the electromagnetic energy density at a generic point of a dielectric planar structure, BSWs are often preferable to modes in which light is confined uniquely by total internal reflec…
This work concentrates on numerical simulations of Photonic Crystal structures using basis-expansion methods, with a main focus on simulating disorder. The plane-wave and guided-mode expansions are outlined as tools to compute the Bloch modes of a structure, on the basis of which the Bloch-mode expansion formalism is outlined - the latter allowing for simulations of large structures in presence of disorder. As a first illustration of the method, we apply it to three gentle-confinement cavities, to obtain results for their quality factors similar to the theoretically predicted in the literature…
Maxwell equations are solved in a layer comprising a finite number of homogeneous isotropic dielectric regions ended by anisotropic perfectly matched layers (PMLs). The boundary-value problem is solved and the dispersion relation inside the PML is derived. The general expression of the eigenvalues equation for an arbitrary number of regions in each layer is obtained, and both polarization modes are considered. The modal functions of a single layer ended by PMLs are found, and their orthogonality relation is derived. The present method is useful to simulate scattering problems from dielectric o…
In this letter, we introduce stacked fishnet metamaterial for steering light in microwave region. We numerically demonstrate that optical Bloch oscillations and a focus of as small as one sixth of a wavelength can be achieved. The flexibility of varying geometrical parameters of the fishnet slabs provides an efficient way for tuning its local effective media parameters and opens the possibility for controlling light arbitrarily. The experiment verifies subwavelength-sized light focusing effect by scanning magnetic field at the surface of the sample directly.
In a lossless periodic structure, a bound state in the continuum (BIC) is characterized by a real frequency and a real Bloch wavevector for which there exist waves propagating to or from infinity in the surrounding media. For applications, it is important to analyze the high-$Q$ resonances that either exist naturally for wavevectors near that of the BIC or appear when the structure is perturbed. Existing theories provide quantitative results for the complex frequency (and the $Q$-factor) of resonant modes that appear/exist due to structural perturbations or wavevector variations. When a period…
We consider propagating bound states in the continuum in dielectric gratings. The gratings consist of a slab with ridges periodically arranged ether on top or on the both sides of the slab. Based on the Fourier modal approach we recover the leaky zones above the line of light to identify the geometries of the gratings supporting Bloch bound states propagating in the direction perpendicular to the ridges. Most importantly, it is demonstrated that if a two-side grating possesses either mirror or glide symmetry the Bloch bound states are stable to variation of parameters as far as the above symme…
A uniform or periodic dielectric slab can serve as an optical waveguide for which guided modes are important, and it can also be used as a diffraction structure for which resonant modes with complex frequencies are relevant. Guided modes are normally studied below the lightline where they exist continuously and emerge from points on the lightline, but isolated guided modes may exist above the lightline and they are the so-called bound states in the continuum. Resonant modes are usually studied above the lightline (defined using the real part of the complex frequency), but they are not connecte…
We recently reported on the first realization of ultraviolet photonic crystal laser based on zinc oxide [Appl. Phys. Lett. {\bf 85}, 3657 (2004)]. Here we present the details of structural design and its optimization. We develop a computational super-cell technique, that allows a straightforward calculation of the photonic band structure of ZnO photonic crystal slab on sapphire substrate. We find that despite of small index contrast between the substrate and the photonic layer, the low order eigenmodes have predominantly transverse-electric (TE) or transverse-magnetic (TM) polarization. Becaus…
We show that a very simple structure consisting of a single subwavelength dielectric ridge on the surface of a slab waveguide enables spatial differentiation and integration of the profile of optical beams propagating in the waveguide. The differentiation and integration operations are performed in transmission and in reflection, respectively, at oblique incidence of the beam impinging on the ridge. The implementation of these operations is associated with the resonant excitation of a cross-polarized eigenmode of the ridge. At the considered parameters of the structure and incidence geometry, …
We studied the optical properties of a dielectric photonic crystal structure with spirals arranged in a hexagonal lattice. The dielectric constant of the material is 9 and the filling ratio is 15.2%. We found that this kind of structure exhibits a significant polarization gap for light incident along the axis of the spirals. The eigenmodes inside the polarization gap are right-hand (left-hand) circularly polarized depending on the whether the spirals are left-handed (right-handed). The transmission spectrum of a slab of such a structure has been calculated and matches well with the analysis of…
We study the eigenmodes of a slab of a wire metamaterial and demonstrate that such a waveguiding structure supports deep-subwavelength propagating modes exhibiting properties of a single-mode waveguide at any fixed frequency below the plasma frequency of metal wires. We compare our analytical results with the dispersion relations extracted from the experimental measurements.
Floquet-Bloch lattices are systems in which wave packets are subjet to periodic modulations both in time and space, showing rich dynamics. While this type of lattices is difficult to implement in solid-state physics, optical systems have provided excellent platforms to probe their physics: among other effects, they have revealed genuine phenomena such as the anomalous Floquet topological insulator and the funnelling of light into localised interface modes. Despite the crucial importance of the band dispersion in the photon dynamics and the topological properties of the lattice, the direct expe…
We present a systematic numerical approach to compute the eigenmodes and the related eigenfrequencies of a disordered photonic crystal, characterized by small fluctuations of the otherwise periodic dielectric profile. The field eigenmodes are expanded on the basis of Bloch modes of the corresponding periodic structure, and the resulting eigenvalue problem is diagonalized on a truncated basis including a finite number of Bloch bands. The Bloch-mode expansion is very effective for modeling modes of very extended disordered structures in a given frequency range, as only spectrally close bands mus…
We present a general analysis for finding and characterizing nonlinear exceptional point (EP) lasers above threshold, using steady-state ab-initio Maxwell-Bloch equations. For a system of coupled slabs, we show that a nonlinear EP is obtained for a given ratio between the external pumps in each resonator, and that it is associated with a kink in the output power and lasing frequency, confirming coupled-mode theory predictions. Through numerical linear stability analysis, we confirm that the EP laser can be stable for a large enough inversion relaxation rate. We further show that the EP laser c…
The existence of a frozen mode in a periodic serpentine waveguide with broken longitudinal symmetry is demonstrated numerically. The frozen mode is associated with a stationary inflection point (SIP) of the Bloch dispersion relation, where three Bloch eigenmodes collapse on each other, as it is an exceptional point of order three. The frozen mode regime is characterized by vanishing group velocity and enhanced field amplitude, which can be very attractive in various applications including dispersion engineering, lasers, and delay lines. Useful and simple design equations that lead to realizati…
We study the classical scattering problem of a plane electromagnetic wave incident on the surface of semi-infinite periodic stratified media incorporating anisotropic dielectric layers with special oblique orientation of the anisotropy axes. We demonstrate that an obliquely incident light, upon entering the periodic slab, gets converted into an abnormal grazing mode with huge amplitude and zero normal component of the group velocity. This mode cannot be represented as a superposition of extended and evanescent contributions. Instead, it is related to a general (non-Bloch) Floquet eigenmode wit…
Split Coaxial Cable Medium for Tunable Artificial Dielectrics and Plasmas
School of Physics and Engineering, ITMO University（ITMO大学物理与工程学院）；School of Engineering, New Uzbekistan University（新乌兹别克斯坦大学工程学院）；Department of Physics, Universidad Nacional de Colombia（哥伦比亚国立大学物理系）
We study the propagation of light in a three-dimensional double-periodic Ag/TiO2 multilayer metamaterial composed of coupled plasmonic waveguides operating in the visible and UV spectral range. For these frequencies, light propagation in the plane of the waveguides is described by a negative phase velocity, while for the orthogonal direction light propagation is described by a Bloch wave composed of a large number of harmonics. As a result, the material cannot generally be described by a single phase index: decomposing the Bloch wave into different harmonics we show that for the wavelength ran…
It has recently been predicted that a conical singularity (= Dirac point) in the band structure of a photonic crystal produces an unusual 1/L scaling of the photon flux transmitted through a slab of thickness L. This inverse-linear scaling is unusual, because it is characteristic of radiative transport via diffusion modes through a disordered medium -- while here it appears for propagation of Bloch modes in an ideal crystal without any disorder. We present a quantitative numerical test of the predicted scaling, by calculating the scattering of transverse-electric (TE) modes by a two-dimensiona…
The Single Gyroid, or srs, nanostructure has attracted interest as a circular-polarisation sensitive photonic material. We develop a group theoretical and scattering matrix method, applicable to any photonic crystal with symmetry I432, to demonstrate the remarkable chiral-optical properties of a generalised structure called 8-srs, obtained by intergrowth of eight equal-handed srs nets. Exploiting the presence of four-fold rotations, Bloch modes corresponding to the irreducible representations E- and E+ are identified as the sole and non-interacting transmission channels for right- and left-cir…
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 s…
The dispersion of a three-way waveguide is engineered to exhibit exceptional modal characteristics. Two coupled waveguides with Parity-Time (PT) symmetry have been previously demonstrated to exhibit second order exceptional points of degeneracy (EPDs). In this work, we introduce and investigate a particular class of EPDs, applicable from radio frequency to optical wavelengths, whereby three coupled waveguides satisfy Glide-Time (GT) symmetry to exhibit a third order modal degeneracy with a real-valued wavenumber. GT symmetry involves glide symmetry of lossless/gainless components of the wavegu…
Bound states in the continuum (BICs) and Fano resonances in planar photonic lattices, including metasurfaces and photonic crystal slabs, have been studied extensively in recent years. Typically, the BICs and Fano resonances are associated with the second stop bands open at the second-order Gamma ($Γ$) point. This paper address the fundamental properties of the fourth stop band accompanied by BICs at the third-order $Γ$ point in one-dimensional (1D) leaky-mode photonic lattices. At the fourth stop band, one band edge mode suffers radiation loss, thereby generating a Fano resonance, while the ot…
