Conventional Plane Wave Expansion Method Research Articles

A rasterized plane wave expansion method for complex 2-D phononic crystals

This paper introduces a computer image processing-based method called the rasterized plane wave expansion method (RPWEM). This method improves the conventional plane wave expansion method (CPWEM) and reduces its formula derivation cost in the case of 2-D phononic crystals with complex inclusions. In addition, for 2-D phononic crystals with arbitrary multiphase materials, this method can also simplify the calculation process. By employing the proposed RPWEM, successful calculations of band gaps are conducted for planar phononic crystals with the hexagonal unit cell as well as phononic crystals featuring integrated circuit-like structures. The results further validate the expanded applicability of the RPWEM in comparison to the CPWEM. In the end, a potential application similar to DIC (Digital Image Correlation) for predicting band structures of phononic crystals is proposed, and its feasibility is briefly validated.

A general method for calculation of Faraday rotation and transmittance in two-dimensional magneto-optic photonic crystals by solving vector-Helmholtz equation in anisotropic media

We propose a general method based on the plane wave expansion method for the calculation of the Faraday rotation and transmittance spectrum of the two-dimensional magneto-optic photonic crystal. The method is an extension of the conventional plane wave expansion method by and taking into account all the three components of the electromagnetic fields and considering the permittivity tensor with off-diagonal imaginary components representing the Faraday effect and then solving the vector Helmholtz equation. The formulation is explained in detail from the implementation points of view and the computation cost of the method. In the end, numerical results are presented to verify the accuracy and convergence of the method.

Efficient Analysis of Confined Guided Modes in Phoxonic Crystal Slabs

Today's standard fabrication processes are just capable of manufacturing slab of photonic and phononic crystals, so an efficient method for analysis of these crystals is indispensable. Plane wave expansion (PWE) as a widely used method in studying photonic and phononic (phoxonic) crystals in full three dimensions is not suitable for slab analysis in its standard form, because of convergence and stability issues. Here, we propose a modification to this method which overcomes these limitations. This improved method can be utilized for calculation of both photonic and phononic modes in phoxonic slabs. While in the standard three-dimensional PWE, Fourier series are used to estimate the field dependence across the normal component of the slab, we expand the fields across the third dimension using eigenmodes of a plain unstructured slab. Despite its approximate nature, this approach is observed to be both much faster and more accurate than the conventional PWE method and can give a very accurate estimation of confined propagating modes. As an application example of the proposed method we investigate a non-reciprocal photonic device. This device is a phoxonic slab waveguide which changes modes of optical waves by elastic waves and can be used as an optical insulator or mode converter.

Enhanced plane wave expansion analysis for the band structure of bulk modes in two-dimensional high-contrast solid–solid phononic crystals

Plane wave expansion analyses that use the inverse rule to obtain the Fourier coefficients of the elastic tensor instead of the more conventional Laurent's rule, exhibit faster convergence rates for solid–solid phononic crystals. In this work, the band structure convergence of calculations using the inverse rule is investigated and applied to the case of high acoustic impedance contrast solid–solid phononic crystals, previously known for convergence difficulties. Results are contrasted to those obtained with the conventional plane wave expansion method. The inverse rule is found to converge at a much rate for all ranges of impedance contrast, and the ratio between the computational times needed to obtain a convergent band structure for a high-contrast solid–solid phononic crystal with the conventional plane wave expansion method using 1369 reciprocal lattice vectors is as large as 6800:1. This ratio decreases for material sets with lower impedance contrast; however, the inverse rule is still faster for a given error threshold for even the lowest impedance contrast phononic crystals reported in the literature. This convergence enhancement is a major factor in reconsidering the plane wave expansion method as an important tool in obtaining propagating elastic modes in phononic crystals.

Analysis of surface modes in photonic crystals by a plane-wave transfer-matrix method

We have developed a plane-wave transfer-matrix method (PWTMM) with the aid of the interpolation technique to analyze the dispersion relation of surface modes in photonic crystal or photonic crystal surface waveguide. The proposed approach has been applied to several surface structures in two-dimensional photonic crystals. The calculated dispersion relation of the surface modes is in good agreement with the result obtained by the conventional plane-wave expansion method in combination with the supercell technique. The developed PWTMM needs to handle only a single unit-cell layer domain and is therefore numerically friendly. The proposed approach can become an efficient and accurate numerical tool to understand and design surface modes in different two-dimensional and three-dimensional photonic crystals with complex geometries.

Solution of photonic band diagrams by a plane-wave-based transfer-matrix method in combination with an interpolation method

We have developed a plane-wave transfer-matrix method in combination with an interpolation method to calculate the whole diagram of photonic band structures in the first Brillouin zone by only looking at one single stacking direction of crystalline layers with fast numerical convergence. Using the interpolation method, one does not need to scan a considerable frequency range to accurately compute the exact value of eigenfrequency corresponding to a Bloch wave. The calculated photonic band structures by our proposed method are in good agreement with results obtained by means of the conventional plane-wave expansion method but with a better numerical convergency. The proposed approach can become an economical, flexible, and accurate numerical tool to understand and design different two-dimensional and three-dimensional photonic crystals.

A plane-wave-based approach for complex photonic band structure and its applications to semi-infinite and finite system

A plane-wave-based approach for complex photonic band structure is developed and applied to analyse the behaviour of electromagnetic (EM) waves at the surface of semi-infinite and finite photonic crystals (PCs). The complex photonic band structures of two-dimensional and three-dimensional PCs have been calculated by the present approach and verified by conventional plane wave expansion method and plane-wave-based transfer matrix method. Based on the eigen modes of present approach, equations in matrix form are given to describe the boundary conditions of PCs, and consequently the behaviour of EM waves at the surface of semi-infinite and finite PCs is discussed. These results have also verified the validity and efficiency of the current approach.

Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals

In this paper, we extend the conventional plane wave expansion method in 3D anisotropic photonic crystal to be able to calculate the complex $\mathbf{k}$ even if permittivity and permeability are complex numbers or the functions of $\omega$. There are some tricks in the derivation process, so we show the process in detail. Besides, we also provide an example for testing and explaining, and we also compare the results with the band structure derived from conventional plane wave expansion method, then we finally find that there is a good consistency between them.

Efficient design and optimization of photonic crystal waveguides and couplers: The Interface Diffraction Method

We present the interface diffraction method (IDM), an efficient technique to determine the response of planar photonic crystal waveguides and couplers containing arbitrary defects. Field profiles in separate regions of a structure are represented through two contrasting approaches: the plane wave expansion method in the cladding and a scattering matrix method in the core. These results are combined through boundary conditions at the interface between regions to model fully a device. In the IDM, the relevant interface properties of individual device elements can be obtained from unit cell computations, stored, and later combined with other elements as needed, resulting in calculations that are over an order of magnitude faster than supercell simulation techniques. Dispersion relations for photonic crystal waveguides obtained through the IDM agree with the conventional plane wave expansion method to within 2.2% of the stopband width.

Application of structural symmetries in the plane-wave-based transfer-matrix method for three-dimensional photonic crystal waveguides

The plane-wave-based transfer-matrix method (TMM) exhibits a peculiar advantage of being capable of solving eigenmodes involved in an infinite photonic crystal and electromagnetic (EM) wave propagation in finite photonic crystal slabs or even semi-infinite photonic crystal structures within the same theoretical framework. In addition, this theoretical approach can achieve much improved numerical convergency in solution of photonic band structures than the conventional plane-wave expansion method. In this paper we employ this TMM in combination with a supercell technique to handle two important kinds of three-dimensional (3D) photonic crystal waveguide structures. The first one is waveguides created in a 3D layer-by-layer photonic crystal that possesses a complete band gap, the other more popular one is waveguides built in a two-dimensional photonic crystal slab. These waveguides usually have mirror-reflection symmetries in one or two directions perpendicular to their axis. We have taken advantage of these structural symmetries to reduce the numerical burden of the TMM solution of the guided modes. The solution to the EM problems under these mirror-reflection symmetries in both the real space and the plane-wave space is discussed in a systematic way and in great detail. Both the periodic boundary condition and the absorbing boundary condition are employed to investigate structures with or without complete 3D optical confinement. The fact that the EM field components investigated in the TMM are collinear with the symmetric axes of the waveguide brings great convenience and clarity in exploring the eigenmode symmetry in both the real space and the plane-wave space. The classification of symmetry involved in the guided modes can help people to better understand the coupling of the photonic crystal waveguides with external channels such as dielectric slab or wire waveguides.

Photonic band structures solved by a plane-wave-based transfer-matrix method.

Transfer-matrix methods adopting a plane-wave basis have been routinely used to calculate the scattering of electromagnetic waves by general multilayer gratings and photonic crystal slabs. In this paper we show that this technique, when combined with Bloch's theorem, can be extended to solve the photonic band structure for 2D and 3D photonic crystal structures. Three different eigensolution schemes to solve the traditional band diagrams along high-symmetry lines in the first Brillouin zone of the crystal are discussed. Optimal rules for the Fourier expansion over the dielectric function and electromagnetic fields with discontinuities occurring at the boundary of different material domains have been employed to accelerate the convergence of numerical computation. Application of this method to an important class of 3D layer-by-layer photonic crystals reveals the superior convergency of this different approach over the conventional plane-wave expansion method.

A finite-difference eigenvalue algorithm for calculating the band structure of a photonic crystal

A new method based on a finite difference of the governing differential equation for the eigenvalue problem is introduced to calculate the band structure of a two-dimensional photonic crystal. The effective medium technique is also used in the method. The problem is reduced to a standard matrix eigenvalue problem. Compared to the conventional plane wave expansion method, the present method improves the convergence of the solution and thus is a fast and accurate algorithm for calculating the band structure of a photonic crystal.

Obtaining the Band Structure of a Two-Dimensional Photonic Crystal by an Averaged Field Approach

An averaged field approach is suggested for obtaining the band structure of a photonic crystal when the dimension of the inclusions is much smaller compared with the longest period of the rectangular lattice of the photonic crystal. The method is illustrated for the H-polarization of the 2-dimensional case. The band structure is obtained in an explicit and simple way. The method is verified numerically by comparing with the conventional plane-wave expansion method.

An averaged-field approach for obtaining the band structure of a dielectric photonic crystal

An averaged-field approach is suggested for obtaining the explicit form of the dispersion equation for electromagnetic waves propagating in a three- or two-dimensional dielectric photonic crystal when the dimension of the inclusions is small compared with the longest period of the rectangular lattice of the photonic crystal. The band structure is obtained in an explicit and simple way. The method is verified numerically by comparing with the conventional plane-wave expansion method for a special case.