Cutoff frequencies in planar optical waveguides with arbitrary index profiles: An efficient numerical method

A numerical method for obtaining mode cutoffs for planar waveguides with arbitrary index profiles is developed. The method is based on the Galerkin method in which the wave equation for modes at cutoff is converted to a matrix eigenvalue equation using a set of orthogonal basis functions. Due to different boundary conditions, we have identified two separate cases; one, in which the cover and the substrate indices are identical leading to same behavior of the field at cutoff in these two regions and, the other, in which the two indices are different and hence, the field behaves differently. We have accordingly chosen different basis functions for the two cases. The method results in a generalized matrix eigenvalue problem which has been converted to a standard symmetric matrix eigenvalue analytically. The method has been used to obtain mode cutoffs for waveguides with a variety of index profiles. Comparisons with available exact results show that very good accuracies can be obtained with moderate matrix sizes.     

Analytical investigations of planar optical waveguides with arbitrary index profile

We present here an improvement of the existing equivalent attenuated vector (EAV) method. It is shown that our study gives not only analytical dispersion equation with clear physical insight, but also the extremely accurate propagation constants for planar waveguides with arbitrary index profiles.     

Analysis of arbitrary index profile planar optical waveguides and multilayer nonlinear structures: a simple finite difference algorithm

we present here a simple numerical method to obtain the mode effective indices as well as field distributions of modes of any arbitrary profile planar optical waveguide. The method is based on the solutions of scalar and semivectorial Helmoltz’s equation by finite difference algorithm and devised with a field convergence technique. This approach is quite general and can be applied straightforwardly to calculate the guided as well as quasi- or leaky modes of any arbitrary planar structure without the need to solve any eigenvalue equation or any complex matrix formalism. Besides providing the ease of application, the algorithm is particularly useful for waveguides with any graded index profile or with irregular multilayered structure and multilayered waveguides with a localised arbitrary nonlinear film. The performance of our method is verified against typical problems with analytical solutions or having results known otherwise, and is shown to yield results with good accuracy.     

A search-and-track algorithm for controlling the number of guided modes of planar optical waveguides with arbitrary refractive index profiles

A search-and-track algorithm is proposed for controlling the number of guided modes of planar optical waveguides with arbitrary refractive index profiles. The algorithm starts with an initial guess point in the parameter space that supports a specific number of guided modes. Then, it searches for, and tracks, the boundaries of this space or another space supporting different number of modes. It does so by monitoring the sign of a unified cutoff dispersion function. The algorithm is applied to both symmetric and asymmetric silicon-based parabolic-index waveguides. It shows that unlike asymmetric waveguides, the single-mode condition of symmetric waveguides is controlled by TM-, as opposed to TE-, polarization. This abnormal polarization control is strongest for high index contrast waveguides of sub-micrometer core sizes. The results are verified by full-vectorial beam propagation method.     

Construction of the refractive index profiles for few-mode planar optical waveguides

We present a nondestructive technique to predict the refractive index profiles of isotropic planar waveguides, on which a thin gold film is deposited to as the cladding. The negative dielectric constant of the metal results in significant differences of effective indices between TE and TM modes. The two polarized modes and a surface plasmon resonance (SPR) with abundant information of the surface index can be used to construct the refractive index profiles of single-mode and two-mode waveguides at a fixed wavelength.     

Simple high-order Galerkin finite element scheme for the investigation of both guided and leaky modes in anisotropic planar waveguides

A simple high-order Galerkin finite element scheme is formulated to compute both the guided and leaky modes of anisotropic planar waveguides with a diagonal permittivity tensor. Transparent boundary conditions derived from the Sommerfield radiation conditions are used to model the fields at the computational boundaries that allow the radiation into the high index cladding/substrate and decay into the low index cladding/substrate, hence work for both guided and leaky modes. Richardson's extrapolation is employed to achieve high-order accuracy by only using simple first-order-polynomial basis functions. Schemes up to sixth-order of accuracy in the effective index are demonstrated. The resulted non-linear sparse matrix eigenvalue equation is solved using an iterative procedure. The ability of the scheme to compute leaky and guided modes of various structures with isotropic and anisotropic materials, step and graded index profiles is demonstrated; including its applications to investigate the properties of ARROW structures.     

A quasi-vector finite difference mode solver for optical waveguides with step-index profiles

A finite difference scheme based on the polynomial interpolation is constructed to solve the quasi-vector equations for optical waveguides with step-index profiles. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are taken into account. The numerical results include the polarization effects, but the memory requirement is the same as in solving the scalar wave equation. Moreover, the proposed finite difference scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for a buried rectangular waveguide and a rib waveguide are presented. Solutions are compared favorably with those obtained by the numerical approaches published earlier.     

Accuracy of three-point finite difference approximations for optical waveguides with step-wise refractive index discontinuities

A rigorous truncation error analysis of three-point finite difference approximations for optical waveguides with step-wise refractive index discontinuities is given. As the basis for the analysis we use the exact coefficients of the series that expresses the field value at a given finite difference node in terms of the field value and its derivatives at a neighbouring node. This series is applied to develop a rigorous formalism for the truncation error analysis of the three-point finite difference approximations used in the numerical modelling of light propagation in optical waveguides with step-wise discontinuities of the refractive index profile. The results show that the approximations reach O(h²) truncation error only asymptotically for sufficiently small values of the mesh size.     

Arbitrary truncation order three-point finite difference method for optical waveguides with stepwise refractive index discontinuities

An exact finite difference (FD) representation of the second-order derivative on three nodes is presented and used to obtain an FD algorithm that allows achieving an arbitrary truncation order. The FD weights are calculated analytically using the series that expresses the field value at a given FD node in terms of the field value and its derivatives at a neighboring node, when a stepwise discontinuity in the refractive index distribution is present between the nodes. The results obtained confirm that the proposed algorithm is accurate, efficient, and achieves the predicted improved performance.     

Reconstruction of extraordinary refractive index profiles of optical planar waveguides with single or double modes fabricated by O ion implantation into lithium niobate

A method named intensity calculation method (ICM), which is based on beam propagation method (BPM) and image processing, was carried out to reconstruct the extraordinary refractive index profile (RIP) of single-mode planar waveguide in lithium niobate (LiNbO₃), which was fabricated by multi-energy megaelectron-volt (MeV) O²⁺ ion implantation. In addition, it has been proved reasonable that the alternation of extraordinary refractive index induced by ion implantation into LiNbO₃ is mainly due to the degradation of polarization and reduction of material physical density. As a result, the possible extraordinary RIP of the double-mode planar waveguide could be reconstructed using BPM according to such a hypothesis and the calculated guiding mode values. The end-fire coupling and m-line arrangements were carried out to obtain the near-field modal patterns and dark-mode spectra of waveguides, respectively.     

Numerical calculations of modes of optical waveguides with two-dimensional refractive index profiles by a field correction method

Abstact We describe a numerical method devised to calculate the lower-order modes of optical waveguides with two-dimensional transverse index profiles. This method is based on finite-difference approximation of the scalar wave equation and the minimization of the error in the field shape of a mode. The field correction method can be applied to waveguides with large index steps and generates the mode's scalar propagation constant and its field. To test the field correction method, it is applied to a three-layer optical fibre and the results are compared with exact solutions. The error in the calculations of the modal effective indices is found to be of the order of 10^-6. Example mode calculations are given for fused couplers and rib waveguides.     

Low-loss waveguides written with a femtosecond laser for flexible interconnection in a planar light-wave circuit

We describe a low-loss single-mode waveguide in planar light-wave circuit (PLC) glass doped with boron and phosphorus, which is more difficult to write than pure-silica glass. The written waveguide has a rectangular core, a symmetric near-field pattern, and a propagation loss of 0.35 dB/cm. The loss that originates from the mode-field mismatch between the mode-field diameters of the written and the PLC waveguides is less than 0.1 dB/point. In addition, we successfully connected PLC waveguides with a 500-microm-long waveguide written with a laser. The laser-written waveguide can flexibly connect PLC waveguides with a low coupling loss.     

Fast quantum search algorithm for databases of arbitrary size and its implementation in a cavity QED system

We propose a method for implementing the Grover search algorithm directly in a database containing any number of items based on multi-level systems. Compared with the searching procedure in the database with qubits encoding, our modified algorithm needs fewer iteration steps to find the marked item and uses the carriers of the information more economically. Furthermore, we illustrate how to realize our idea in cavity QED using Zeeman?s level structure of atoms. And the numerical simulation under the influence of the cavity and atom decays shows that the scheme could be achieved efficiently within current state-of-the-art technology.     

Radiative heat transfer in a planar medium with anisotropic scattering and directional boundaries

Radiation heat transfer in an absorbing, emitting and scattering medium has been the subject of many previous investigations. Most solutions are numerically complex and the existing analytical solutions are restricted in application by the simplifying assumptions involved. A plane-parallel medium is considered which scatters anisotropically. The boundaries are considered to be specular reflectors, as predicted by Fresnel's relations, while the diffusely incident radiation is refracted according to Snell's law. The emission is restricted to a medium with a uniform temperature distribution. Approximate closed-form solutions for the radiative heat flux and incident intensity are presented for dielectric layers and linear anisotropic scattering. Numerical results are also presented and show that the effects of directional boundaries, anisotropic scattering, scattering albedo and optical depth are accurately predicted by the approximate solution.     

The T-matrix for particle with arbitrary permittivity tensor and parallelization of the computational code

Using the T-matrix approach electromagnetic scattering by a non-axisymmetric particle with an arbitrary permittivity tensor is studied. The electromagnetic fields inside the scatterer are expressed by a system of quasi-spherical vector wave functions which are derived by the use of inverse Fourier transform. Using this expansion a solution of the light scattering problem in the framework of the null-field method is obtained. The implemented T-matrix program is parallelized using both OpenMP and MPI parallel paradigms. Numerical scattering results for anisotropic ellipsoids are presented.     

Angular tolerance and sensitivity for designing Fresnel-like prism structures with arbitrary surface profiles

Fresnel-like prism arrays are designed using geometric ray optics and fabricated on a curved surface according to specific focal length requirements. In this study, the tolerance of Fresnel-like prism structures on arbitrary surfaces was investigated. Compact equations (iterative and non-iterative) based on analytical formulation can be helpful in exploring the sensitivity associated with the fabrication of Fresnel-like structures. The curved surface in this study revealed a linear relationship between maximum angular tolerance and the angle of incident rays. An understanding of such linearity is a key issue in the fabrication of lenses as well as in the sensitivity formulae used for optical structures in the development of next-generation optics.     

平面光波导用于实时测试生化反应新方法的研究

在分析经典消逝波光波导生化传感器的基础上,为寻找性能更佳的传感器,探讨用更为直观的截止特性来进行生化传感。分别对三层及四层平板波导进行了研究,指出了作为基于截止特性的生化传感器三层波导结构存在的局限性。在理论上研究了可以在四层波导中使用变折射率材料来进行基于截止特性的生化传感,并给出了由空气隙、极化聚合物、高折射率薄膜、待测物组成的传感器结构示意图。用数值分析的方法绘出了此装置的灵敏度与高折射率薄膜厚度的关系曲线,并将它同经典消逝波光波导生化传感器灵敏度与薄膜厚度关系图进行了比较。结果表明,这种新型传感器理论上可以在较厚的薄膜结构中达到很高的灵敏度。