Loss and dispersion tailoring in 1D photonics band gap Bragg reflection waveguides: finite chirped claddings as a design tool

We exploit a simple and accurate matrix method to analyze the effects of introducing a linear chirp either in thickness or in refractive index of the cladding layers on the propagation characteristics (loss and dispersion) of 1D photonic band gap planar Bragg reflection waveguides (BRWs). We show that an appropriate chirp in the otherwise periodic claddings of finite extent BRWs could be gainfully exploited to tailor its leakage loss and waveguide dispersion features. In particular, we theoretically demonstrate that for some reported sample BRWs, leakage loss and waveguide dispersion could be significantly reduced by a factor of 30–50 and by about two orders of magnitude, respectively as compared to un-chirped BRWs. Furthermore, we also show that in contrast to un-chirped BRWs, how chirped BRWs could be designed with attractive feature like much less number of cladding layers and nearly wavelength independent losses. Our analysis and proposal should serve as a useful design tool to tailor the propagation characteristics of BRWs.     

Spatial harmonics modelling of planar periodic segmented waveguides

A theoretical approach based on Bloch theorem and spatial harmonics has been used to investigate the propagation characteristics in planar periodic segmented waveguides. This analytical method allows to evaluate the field distribution, the effective index and the propagation losses of these structures, taking into account both the propagating and the counterpropagating field components, thus evidencing all the phenomena which can take place in: Bragg reflections, leaky waves, etc. Results for TE fields are presented and compared with those obtained using a paraxial 2D FD-BPM method and a Padé based one, showing that no more than (1, 1) order approximants are needed to provide good estimates of the device characteristics.     

Plane wave propagation and frequency band gaps in periodic plates and cylindrical shells with and without heavy fluid loading

Free plane wave propagation in infinitely long periodic elastic structures with and without heavy fluid loading is considered. The structures comprise continuous elements of two different types connected in an alternating sequence. In the absence of fluid loading, an exact solution which describes wave motion in each unboundedly extended element is obtained analytically as a superposition of all propagating and evanescent waves, continuity conditions at the interfaces between elements are formulated and standard Floquet theory is applied to set up a characteristic determinant. An efficient algorithm to compute Bloch parameters (propagation constants) as a function of the excitation frequency is suggested and the location of band gaps is studied as a function of non-dimensional parameters of the structure's composition. In the case of heavy fluid loading, an infinitely large number of propagating or evanescent waves exist in each unboundedly extended elasto-acoustic element of a periodic structure. Wave motion in each element is then presented in the form of a modal decomposition with a finite number of terms retained in these expansions and the accuracy of such an approximation is assessed. A generalized algorithm is used to compute Bloch parameters for a periodic structure with heavy fluid loading as a function of the excitation frequency and, similarly to the previous case, the location of band gaps is studied.     

非线性薄膜波导TE模色散特性的多层分割法计算

对于芯区为非线性介质、衬底及包层为线性介质的平板波导、提出用多层分割法分析芯区的模场,采用递推公式（等效于推广的传递矩阵法）求解传播常数与光功率间的依赖关系,该法适用地克尔型或非克尔型介质及芯区折射率非均匀的一般情形,实例计算结果与已有的精确数值计算结果十分符合。     

Dispersion Characteristics of Arbitrary Periodic Structures with Rectangular Grooves

The dispersion characteristics of a circular cylindrical waveguide with periodic surface corrugations consisting of rectangular grooves with smoothing are examined using the Space Harmonic Method (SHM). The whole structure is divided into two regions, one describing the propagation volume and one inside the grooves. In the first region, the Floquet theorem is applicable and the field distribution is expressed as a summation of spatial Bloch components, whereas in the second one an appropriate Fourier expansion of standing waves is used. Applying the boundary conditions an infinite system of equations is obtained, which is solved numerically by truncation. Several cases are considered, including the limiting cases of a sinusoidal and a rectangular corrugation profile, to check the accuracy of the method proposed as well as its dependence on the corrugation profile. Numerical results are presented only for transverse magnetic modes, although the formalism can be easily extended to include all kinds of waves that can in principle propagate in such a structure.     

Single-mode propagation in triangular tube lattice hollow-core terahertz fibers

The mode properties of hollow-core fibers with a cladding formed by a periodic arrangement of dielectric tubes in a triangular lattice for THz applications are numerically investigated. The fiber supports a high number of modes. Effectively single-mode operation can be obtain by reducing the core size, but at the expense of high fundamental mode propagation loss. Single-mode propagation can be obtained by exciting the fiber with a linear polarized gaussian beam with proper spot size.     

COUPLED WAVES ON A PERIODICALLY SUPPORTED TIMOSHENKO BEAM

A mathematical model is presented for the propagation of structural waves on an infinitely long, periodically supported Timoshenko beam. The wave types that can exist on the beam are bending waves with displacements in the horizontal and vertical directions, compressional waves and torsional waves. These waves are affected by the periodic supports in two ways: their dispersion relation spectra show passing and stopping bands, and coupling of the different wave types tends to occur. The model in this paper could represent a railway track where the beam represents the rail and an appropriately chosen support type represents the pad/sleeper/ballast system of a railway track. Hamilton's principle is used to calculate the Green function matrix of the free Timoshenko beam without supports. The supports are incorporated into the model by combining the Green function matrix with the superposition principle. Bloch's theorem is applied to describe the periodicity of the supports. This leads to polynomials with several solutions for the Bloch wave number. These solutions are obtained numerically for different combinations of wave types. Two support types are examined in detail: mass supports and spring supports. More complex support types, such as mass/spring systems, can be incorporated easily into the model.     

Wave dispersion in periodic post-buckled structures

Wave propagation in pinned-supported, post-buckled beams can be described with the Korteweg de Vries (KdV) equation. Finite-element simulations however show that the KdV is applicable only to post-buckled beams with strong pre-compression. For weak and moderate pre-stress, a dispersive front is present and it is the aim of the current paper to analyze sources of dispersion beyond periodicity given three support types: guided, pinned, and free. Bloch theorem and a transfer-matrix method are employed to obtain numerical dispersion relations and characteristic wave modes, which are used to analyze the effects of pre-stress, initial curvature, and the influence of support types. Additionally, a new method is proposed to obtain a semi-analytical dispersion equation for the acoustic branch. Powers of frequency and the propagation constant are explicitly expressed and their coefficients are based on stiffness and mass-matrix components obtained from finite elements. This allows a physical interpretation of the dispersion sources, based on which, equivalent mass–spring models of post-buckled beam are proposed. It is found that mass and stiffness coupling are significant dispersion sources. In the present paper, a reduced form of Bloch theorem is presented exploiting glide-reflection symmetries, reducing the size of the unit cell and allowing an easier representation and interpretation of results.     

Limitations in the perturbation analysis of bent finite-clad fibres and waveguides

The transverse shift in the field distribution and the correction to the propagation constant of the fundamental and symmetric cladding modes on bent finite-clad single-mode fibres and slab waveguides are evaluated from perturbation theory for effective index values extending below the cladding index. Analytical results are derived in both geometries for the step-profile that are valid within the overall limitations of the theory. However, it is found that, for the fibre geometry only, the method breaks down at certain discrete wavelengths because of degeneracies that occur between the HE₁₂ (LP₀₂) and TE₀₁ (LP₁₁) mode propagation constants.     

Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices

This paper presents an exact, wave-based approach for determining Bloch waves in two-dimensional periodic lattices. This is in contrast to existing methods which employ approximate approaches (e.g., finite difference, Ritz, finite element, or plane wave expansion methods) to compute Bloch waves in general two-dimensional lattices. The analysis combines the recently introduced wave-based vibration analysis technique with specialized Bloch boundary conditions developed herein. Timoshenko beams with axial extension are used in modeling the lattice members. The Bloch boundary conditions incorporate a propagation constant capturing Bloch wave propagation in a single direction, but applied to all wave directions propagating in the lattice members. This results in a unique and properly posed Bloch analysis. Results are generated for the simple problem of a periodic bi-material beam, and then for the more complex examples of square, diamond, and hexagonal honeycomb lattices. The bi-material beam clearly introduces the concepts, but also allows the Bloch wave mode to be explored using insight from the technique. The square, diamond, and hexagonal honeycomb lattices illustrate application of the developed technique to two-dimensional periodic lattices, and allow comparison to a finite element approach. Differences are noted in the predicted dispersion curves, and therefore band gaps, which are attributed to the exact procedure more-faithfully modeling the finite nature of lattice connection points. The exact method also differs from approximate methods in that the same number of solution degrees of freedom is needed to resolve low frequency, and arbitrarily high frequency, dispersion branches. These advantageous features may make the method attractive to researchers studying dispersion characteristics, band gap behavior, and energy propagation in two-dimensional periodic lattices.     

Theoretical and experimental investigations of flexural wave propagation in periodic grid structures designed with the idea of phononic crystals

The propagation characteristics of flexural waves in periodic grid structures designed with the idea of phononic crystals are investigated by combining the Bloch theorem with the finite element method. This combined analysis yields phase constant surfaces, which predict the location and the extension of band gaps, as well as the directions and the regions of wave propagation at assigned frequencies. The predictions are validated by computation and experimental analysis of the harmonic responses of a finite structure with 11× 11 unit cells. The flexural wave is localized at the point of excitation in band gaps, while the directional behaviour occurs at particular frequencies in pass bands. These studies provide guidelines to designing periodic structures for vibration attenuation.     

周期势场中电子的中心方程及其应用

周期势场中电子的薛定谔微分方程变换为K空间的称为中心方程的线性齐次方程组，利用此方程可以证明布洛赫定理，讨论弱周期势场中电子的能带。     

Femtosecond laser micromachined optical waveguides in LiTaO3 crystal

By using femtosecond laser micromachining, optical wave guides in both depressed cladding and dual‐line configurations have been produced in LiTaO₃ crystal. The guiding properties and the thermal stability have been investigated for both geometries, which exhibit different performance. Depressed cladding waveguides support guidance along both extraordinary and ordinary index polarizations, while dual‐line waveguides support only extraordinary index polarization. Thermal annealing has been proved to be an effective method to reduce the propagation losses. For the cladding waveguide, the lowest propagation loss was as low as 0.38 dB/cm after the annealing treatment at 400 °C. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Propagation of acoustic waves in the woodpile sonic crystal with a defect

Acoustic wave propagation in a woodpile sonic crystal with a defect is studied theoretically and experimentally. The woodpile sonic crystal is composed of polymethyl methacrylate square rods which orthogonally stacked together, and it is embedded in air background. Defects are created by varying the width and positions of the middle rods in the periodic structure. Defect bands and transmission spectra are calculated by using the finite element method with the periodic boundary condition and the Bloch–Floquet theorem. Frequencies of defect bands are strongly dependent on the width and positions of the middle rods in the periodic structure. The experimental transmission spectra of the woodpile sonic crystals with a defect are also presented and compared with the numerical results. The defect mode properties of the woodpile sonic crystal with a defect can be applied to design novel acoustic devices for filtering sound and trapping sound in defects.     

Electromagnetic waves in uniaxial anisotropic chiral waveguides in magnetized plasma

The characteristics of guided modes in circular waveguides of a uniaxial anisotropic chiral core and a cladding filled with anisotropic plasma are presented. The cladding region is assumed to be infinitely extended with an external applied magnetic field oriented along the direction of propagation in the waveguide. The characteristics equation for the modes in this waveguide are obtained. The variations of the propagation properties with the plasma parameters, chiral parameters, and the cyclotron frequency of plasma have been investigated. Particularly, the effects of the chirality and the cyclotron frequency of plasma on the magnitude and orientation of the energy flux of the guided modes for three kinds of uniaxial anisotropic chiral media have been numerically investigated. Comparisons of the computed results of the presented formulations with published results for some special cases confirm the accuracy of the presented analyses.     

Numerical Analysis of Multilayer Waveguides Using Effective Refractive Index Method

With the help of the effective refractive index method we have numerically analyzed a multilayer planar waveguide structure and calculated the propagation constants, confinement factors, and transverse electric (TE) modes. A five-layer waveguide model has been provided to analyze the electro-magne tic wave propagation process. The analysis method has been applied to the 980 nm laser with active layer of GaInAs/GaInAsP strained quantum wells, GaInAsP confinement layers and GaInP cap layers. By changing the thickness of confinement layers, we obtained confinement factor as high as 95% with higher TE modes TE1 and TE2. The results are in good agreement with the experiment by A. Al-Muhanna et al. and give the new idea to enhance output power of semiconductor lasers. The analysis method can also be extended to any other slab multilayer waveguide structures, and the results are useful to the fabrication of optic-electronic devices.     

Excitation of defect modes from the extended photonicband-gap structures of 1D photonic lattices

This paper stuides numerically the model equation in a one dimensional defective photonic lattice by modifying the potential function to a periodic function. It is found that defect modes (DMs) can be regarded as Bloch modes which are excited from the extended photonic band-gap structure at Bloch wave-numbers with k x = 0. The DMs for both positive and negative defects are considered in this method.     

Modeling of Lamb wave propagation in plate with two-dimensional phononic crystal layer coated on uniform substrate using plane-wave-expansion method

We show that the conversional three-dimensional plane wave expansion method can be revised to investigate the lamb wave propagation in the plate with two-dimensional phononic crystal layer coated on uniform substrate. We find that an imaginary three-dimensional periodic system can be constructed by stacking the studied plates and vacuum layers alternately, and then the Fourier series expansion can be performed. The difference between our imaginary periodic system and the true three-dimensional one is that, in our system, the Bloch feature of the wave along the thickness direction is broken. Three different systems are investigated by the proposed method as examples. The principle and reliability of the method are also discussed.     

Broadband locally resonant beams containing multiple periodic arrays of attached resonators

The plane wave expansion method is extended to study the flexural wave propagation in locally resonant beams with multiple periodic arrays of attached spring-mass resonators. Complex Bloch wave vectors are calculated to quantify the wave attenuation performance of band gaps. It is shown that a locally resonant beam with multiple arrays of damped resonators can achieve much broader band gaps, at frequencies both below and around the Bragg condition, than a locally resonant beam with only a single array of resonators, although the two systems have the same total resonator masses.     

A full-wave three-dimensional analysis of open dielectric waveguides and periodic structures using an equivalent network model

An accurate fully vectorial wave analysis to calculate the propagation characteristics and field distribution of open dielectric waveguides and optical integrated circuits is presented. Based on the periodic repetition concept and using an equivalent network model, a method to analyze wave propagation in these guides is developed, which is able to take the simultaneous presence of mixed polarizations into account. Compared with conventional approaches like mode matching as well as with other numerical methods, the presented three-dimensional (3D) method shows individual advantages like accuracy, simplicity, and numerical efficiency. To show both validity and usefulness of this approach, some fundamental structures are investigated and the obtained numerical results are presented.