Band Structure Of Photonic Crystals Research Articles

Spectral element method for band structures of two-dimensional anisotropic photonic crystals

A spectral element method (SEM) is proposed for the accurate calculation of band structures of two-dimensional anisotropic photonic crystals. It uses Gauss-Lobatto-Legendre polynomials as the basis functions in the finite-element framework with curvilinear quadrilateral elements. Coordination mapping is introduced to make the curved quadrilateral elements conformal with the problem geometry. Mixed order basis functions are used in the vector SEM for full vector calculation. The numerical convergence speed of the method is investigated with both square and triangular lattices, and with isotropic and in-plane anisotropic media. It is shown that this method has spectral accuracy, i.e., the numerical error decreases exponentially with the order of basis functions. With only four points per wavelength, the SEM can achieve a numerical error smaller than 0.1%. The full vector calculation method can suppress all spurious modes with nonzero eigenvalues, thus making it easy to filter out real modes. It is thus demonstrated that the SEM is an efficient alternative method for accurate determination of band structures of two-dimensional photonic crystals.

Study of Defect Modes in Photonic Band Structure of Magnetic Photonic Crystals by Means of Green's Function

In this paper, we create an analytical model to investigate the localized defect modes associated with a defect cell inserted into a one-dimensional magnetic photonic crystal. The structure is a magnetic superlattice with alternative layers of two difierent magnetic permeability containing a defect cell which is a layer of difierent nature (material or/and size) from the other layers. This leads to appearance of several localized defect modes within the magnetic photonic band gap. Our analytical approach is based on the transfer matrix and Green’s function methods to calculate the frequency and number of the defect modes which can be controlled easily by varying parameter values of the constituent layers of the magnetic photonic crystal. This method results in an analytical formula for prediction of frequency of the defect modes for both TE and TM polarizations at arbitrary angle of incidence.

Effect of Metallic Slab Cladding on Photonic Crystal Band Structures

Effect of Metallic Slab Cladding on Photonic Crystal Band Structures

Numerical Investigation of a Bidirectional Wave Coupler Based on Plasmonic Bragg Gratings in the Near Infrared Domain

In this paper, we present a theoretical discussion of the design of bidirectional wave couplers based on plasmonic Bragg gratings in the near infrared domain. A key feature in the design of the plasmonic Bragg gratings is the dependence of the effective refractive index on the thickness of the dielectric layer. These gratings, which function as band rejection filters, enable directional coupling of different SPP modes. By placing two gratings with different band gaps on opposite sides of a subwavelength metallic slit, a bidirectional plasmonic surface wave coupler can be realized. Two-dimensional (2-D) FDTD simulations were performed to elucidate the properties of the device, and were found to agree well with the theoretical predictions. Finally, the wave confinement properties of the plasmonic Bragg gratings are studied further by introducing the equivalent 1-D photonic crystal band structures.

Metallic photonic-band-gap filament architectures for optimized incandescent lighting

We identify an optimized three-dimensional metallic photonic-band-gap filament architecture for electrically pumped, quasithermal, visible light emission. This identification is based on extensive band structure and finite-difference time-domain calculations of metallic photonic crystals. The optimum structure consists of an inverse square-spiral photonic crystal, exhibiting a large bandwidth optical passband below the effective plasma screening frequency of the periodically structured metal. Light emission from the interior surfaces of the filament to the interior air channels occurs exclusively into the passband modes, enabling high-efficiency conversion of electrical energy into visible light.

Uncoupled modes and all-angle negative refraction in walled honeycomb photonic crystals

Left-handed materials have superlensing effects that enable them to surmount the optical diffraction limit. A photonic crystal is able to mimic some properties of all-angle left-landed materials. In this study, the all-angle negative refraction criteria of photonic crystals are evaluated. The MIT Photonic-Bands program is employed to calculate the band structure of walled honeycomb photonic crystals, and the finite-difference time-domain method is used to provide a snapshot of the electric field distribution inside and outside the honeycomb photonic crystals. The results indicate that the all-angle negative refraction phenomena of the honeycomb photonic crystals are correlated with the orientation of the photonic crystals. Furthermore, the role of the uncoupled modes varies based on their orientation to the all-angle negative refraction photonic crystals, in one case assisting negative refraction and in the other case preventing it. The results suggest that symmetric properties should not be ignored when considering the negative refraction of photonic crystals.

Proposed method for detection of the pseudospin-12Berry phase in a photonic crystal with a Dirac spectrum

We propose a method to detect the geometric phase produced by the Dirac-type band structure of a triangular-lattice photonic crystal. The spectrum is known to have a conical singularity ($=\text{Dirac}$ point) with a pair of nearly degenerate modes near that singularity described by a spin-$\frac{1}{2}$ degree of freedom $(=\text{pseudospin})$. The geometric Berry phase acquired upon rotation of the pseudospin is in general obscured by a large and unspecified dynamical phase. We use the analogy with graphene to show how complementary media can eliminate the dynamical phase. A transmission minimum results as a direct consequence of the geometric phase shift of $\ensuremath{\pi}$ acquired by rotation of the pseudospin over $360\ifmmode^\circ\else\textdegree\fi{}$ around a perpendicular axis. We support our analytical theory based on the Dirac equation by a numerical solution of the full Maxwell equations.

Computation of the band structure of two-dimensional photonic crystals with hp finite elements

The band structure of 2D photonic crystals – a periodic material with discontinuous dielectrical properties – and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremely fast, i.e. exponentially, when using p-FEM for smooth and hp-FEM for polygonal interfaces and boundaries. In this article, we discretise the variational eigenvalue problems for photonic crystals with smooth and polygonal interfaces in scalar variables with quasi-periodic boundary conditions by means of p- and hp-FEM – this for the transverse electric (TE) and transverse magnetic (TM) modes. Our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties.

Complete photonic bandgaps for all polarizations in one-dimensional photonic crystals

The photonic band structures of one-dimensional photonic crystals (1D PCs) with threelayers in a period are studied theoretically. Our results show that complete photonicbandgaps (PBGs) for all polarizations can be obtained in these 1D PCs made of aconventional dielectric material, a negative-permittivity and a negative-permeabilitymetamaterial. The origin of the complete PBGs lies in the existence of surface waves for allpolarizations.

Tuning the band gap of self-assembled superparamagnetic photonic crystals in colloidal magnetic fluids using external magnetic fields

The model of tunable superparamagnetic photonic crystals self-assembled in colloidal magnetic fluids under externally applied magnetic fields is established. The mechanisms, which are in charge of the tunability of the band gaps with magnetic fields are clarified. The band structures of the triangularly-arrayed two-dimensional photonic crystals with limited heights of magnetic columns are calculated with the experimental data of structures and refractive indices in the literatures. The field-dependent properties of the first band gaps are gained for the z-odd and z-even modes, respectively. Simulation results indicate that the mid frequencies of the first band gaps of the z-odd modes can be easily tuned by the external magnetic fields, while those of the z-even modes bear relatively weak dependence on the external magnetic fields. Simultaneously, the first band gaps of both kinds of modes become wide along with the increase of the magnetic fields. The results presented in this work give a guideline for realizing the tunable photonic crystals with magnetically colloidal materials and magnetic stimuli.

Influence of Kerr nonlinearity on the band structures of two-dimensional photonic crystals

Dynamic modifications of the band structures of two-dimensional photonic crystals composed of Kerr-nonlinear dielectric rods in air forming hexagonal, square and honeycomb lattices are investigated. Calculations are carried out by the method based on the finite-difference time-domain (FDTD) technique for both TM and TE polarizations. The bands in all cases are observed to red-shift with the introduction of nonlinearity. Inspection of the variation of the mid-gap frequencies with source intensity reveals that the red-shift increases as gap number increases, in a manner proportional to the mid-gap frequency in linear regime. The red-shift of mid-gap frequencies is also found to follow the adopted saturable model for the change of relative permittivity with source intensity. Either the so-called dielectric or the air band edges of the gaps are found to be more sensitive to nonlinearity resulting in broadening or shrinking of the gaps. Possible utilizations of the present observations for all-optical device applications, such as wavelength multiplexing, are discussed.

Impurity effects on the band structure of one-dimensional photonic crystals: experiment and theory

We study the effects of single impurities on the transmission in microwave realizations of the photonic Kronig–Penney model, consisting of arrays of Teflon pieces alternating with air spacings in a microwave guide. As only the first propagating mode is considered, the system is essentially one-dimensional (1D) obeying the Helmholtz equation. We derive analytical closed form expressions from which the band structure, frequency of defect modes and band profiles can be determined. These agree very well with experimental data for all types of single defects considered (e.g. interstitial and substitutional) and show that our experimental set-up serves to explore some of the phenomena occurring in more sophisticated experiments. Conversely, based on the understanding provided by our formulae, information about the unknown impurity can be determined by simply observing certain features in the experimental data for the transmission. Further, our results are directly applicable to the closely related quantum 1D Kronig–Penney model.

Impact of lattice-shape moduli on band structure of photonic crystals

We conduct a comprehensive study of the effect of the Bravais lattice-shape moduli on the band structure and in particular the band gap of photonic crystals. Unlike the conventional comparisons between triangular and rectangular photonic crystals, where the effect of the volume modulus is not separated, we rigorously decouple the volume modulus and determine the differences that can be attributed only to the shape of the lattice. We observe that the triangular lattice enjoys the largest band gap owing to its unique symmetry properties. We also show that the band gap decreases when the ratio of the lattice constants differs from unity. The use of an appropriate parametrization of the lattice-shape moduli combined with the inherent scaling invariance in the Maxwell equations allows us to cover Bravais lattices of all shape and volume moduli completely and without redundancy.

Spectroscopy of nonlinear band structures of one-dimensional photonic crystals

The temporal development of extended nonlinear modes of a one-dimensional photonic crystal during the build-up of the nonlinearity is experimentally investigated. For this a prism coupling setup is used, which allows for a direct comparison of linear and nonlinear photonic band structures. The experimental results are compared with numerical calculations which make use of the Floquet-Bloch approach and the finite difference approximation.

An Efficient Approach to Identifying a Complete Photonic Band Gap in Two-dimensional Photonic Crystals with Omnidirectional Light Propagation

Omnidirectional light propagation in two-dimensional (2D) photonic crystals (PCs) is investigated. An e-cient approach to identifying a complete photonic band gap (PBG) in 2D PCs has been developed. The in-plane band structure of 2D photonic crystals is calculated by an adaptive flnite element method. With adopting the suitable boundary conditions, the eigenvalues can be easily and rapidly calculated no matter how complex the geometric structures are. By symmetry, the omnidirectional photon density of states (PDOS) can be calculated based on the in-plane dispersion relation within the irreducible Brillouin zone. The PDOS corresponding to both the radiation and evanescent waves can be obtained accurately and e-ciently. We demonstrate that the \complete band gaps showed by some previous work due to considering only the radiation modes will be closed by including the contributions of the evanescent modes. These results are of general importance and relevant to the spontaneous emission by an atom, or to dipole radiation in two-dimensional periodic structures.

Photonic band structure calculations for metal-clad two-dimensional photonic crystal slab

We study the effect of metallic cladding on the band structure of a two-dimensional photonic crystal slab. Band structures of a honeycomb patterned two-dimensional photonic crystal slab, with/without a metal-clad or a perfect conductor clad, are calculated using both the finite-difference time-domain (FDTD) method and the plane wave expansion (PWE) method. We explain the changes of the band structure due to cladding. We also show that the band structure of a metal-clad photonic crystal can be obtained effectively from the odd modes of the air bridge type dielectric photonic crystal slab, thicker than the metal-clad photonic crystal by a factor larger than two.

Band structure of two-dimensional photonic crystals that include dispersive left-handed materials and dielectrics in the unit cell

We determine the band structure of two-dimensional photonic crystals that are composed of left-handed materials and dielectrics, based on the numerical solution of the Helmholtz equation by using integral equations. It is found that plasmonic resonances appear constituting a band that is independent of the filling fraction. Wide bandgaps are present where the penetration depth of the electromagnetic field inside the photonic crystal is quite short compared to purely dielectric photonic crystals.

Study on temperature property of band structures in one-dimensional photonic crystals

Using transfer matrix method, the optical transmission properties in one-dimensional (1-D) photonic crystal is analyzed. When the temperature varies, not only the refractive index of the optical medium is changed because of the thermo-optical effect, but also the thickness of the optical medium is changed due to the thermal-expansion effect. Thus, the structure of 1/4 wave-plate stack in original photonic crystal is destroyed and the band structure varies. In this work, the effects of the temperature variation on the first and second band gap in a 1-D photonic crystal are analyzed in detail. It is found that the changes of the starting wavelength, the cut-off wavelength and the forbidden band width depend linearly on the temperature.

Far-Field Focus and Dispersionless Anticrossing Bands in Two-Dimensional Photonic Crystals

We review the simulation work for the far-field focus and dispersionless anticrossing bands in two-dimensional (2D) photonic crystals. In a two-dimensional photonic-crystal-based concave lens, the far-field focus of a plane wave is given by the distance between the focusing point and the lens. Strong and good-quality far-field focusing of a transmitted wave, explicitly following the well-known wave-beam negative refraction law, can be achieved. The spatial frequency information of the Bloch mode in multiple Brillouin zones (BZs) is investigated in order to indicate the wave propagation in two different regions. When considering the photonic transmission in a 2D photonic crystal composed of a negative phase-velocity medium (NPVM), it is shown that the dispersionless anticrossing bands are generated by the couplings among the localized surface polaritons of the NPVM rods. The photonic band structures of the NPVM photonic crystals are characterized by a topographical continuous dispersion relationship accompanied by many anticrossing bands.

Investigation of planar photonic crystal band diagrams under the light cone using surface coupling techniques

We report experimental results for the band structure of 2-dimensional triangular photonic crystals etched in SOI wafers. Boundary conditions at the surface and at the border of the etched area of the slab, allow the excitation of guided modes via diffraction phenomena using surface coupling techniques and the determination of the structure band diagram above and below the light cone. These non-destructive techniques, usable for non-active materials, allow fast characterization of technical processes and the location in reciprocal space of areas with unique dispersion properties.